From a778b60d0d923b277667aace3c79bec3360dac15 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 26 Aug 2024 11:07:52 +0000 Subject: [PATCH] build based on 7aeb38b --- dev/.documenter-siteinfo.json | 2 +- dev/dae/fully_implicit/index.html | 6 - dev/dynamical/nystrom/index.html | 24 - dev/dynamical/symplectic/index.html | 6 - dev/explicit/AdamsBashforthMoulton/index.html | 17 + dev/explicit/Extrapolation/index.html | 30 + dev/explicit/Feagin/index.html | 17 + dev/explicit/HighOrderRK/index.html | 26 + dev/explicit/LowOrderRK/index.html | 68 + dev/explicit/LowStorageRK/index.html | 117 ++ dev/explicit/Nordsieck/index.html | 17 + dev/explicit/PRK/index.html | 17 + dev/explicit/QPRK/index.html | 19 + dev/explicit/RKN/index.html | 18 + dev/explicit/SSPRK/index.html | 53 + dev/explicit/SymplecticRK/index.html | 18 + dev/explicit/Tsit5/index.html | 19 + dev/explicit/Verner/index.html | 29 + dev/imex/IMEXMultistep/index.html | 13 + dev/imex/imex_multistep/index.html | 6 - dev/imex/imex_sdirk/index.html | 7 - 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-Methods for Fully Implicit ODEs (DAEProblem) · OrdinaryDiffEq.jl
GitHub

Methods for Fully Implicit ODEs (DAEProblem)

Missing docstring.

Missing docstring for DImplicitEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for DABDF2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for DFBDF. Check Documenter's build log for details.

diff --git a/dev/dynamical/nystrom/index.html b/dev/dynamical/nystrom/index.html deleted file mode 100644 index 3d92ee0e96..0000000000 --- a/dev/dynamical/nystrom/index.html +++ /dev/null @@ -1,24 +0,0 @@ - -Runge-Kutta Nystrom Methods · OrdinaryDiffEq.jl
GitHub

Runge-Kutta Nystrom Methods

OrdinaryDiffEqRKN.IRKN3 — Type
IRKN3

Improved Runge-Kutta-Nyström method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only.

Second order ODE should not depend on the first derivative.

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer} }

source
OrdinaryDiffEqRKN.IRKN4 — Type
IRKN4

Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only.

Second order ODE should not be dependent on the first derivative.

Recommended for smooth problems with expensive functions to evaluate.

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer} }

source
OrdinaryDiffEqRKN.Nystrom4 — Type
Nystrom4

A 4th order explicit Runge-Kutta-Nyström method which can be applied directly on second order ODEs. Can only be used with fixed time steps.

In case the ODE Problem is not dependent on the first derivative consider using Nystrom4VelocityIndependent to increase performance.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.Nystrom4VelocityIndependent — Type
Nystrom4VelocityIdependent

A 4th order explicit Runkge-Kutta-Nyström method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).

More efficient then Nystrom4 on velocity independent problems, since less evaluations are needed.

Fixed time steps only.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.Nystrom5VelocityIndependent — Type
Nystrom5VelocityIndependent

A 5th order explicit Runkge-Kutta-Nyström method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative). Fixed time steps only.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.FineRKN4 — Type
FineRKN4()

A 4th order explicit Runge-Kutta-Nyström method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

References

@article{fine1987low,
-  title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods},
-  author={Fine, Jerry Michael},
-  journal={Computing},
-  volume={38},
-  number={4},
-  pages={281--297},
-  year={1987},
-  publisher={Springer}
-}
source
OrdinaryDiffEqRKN.FineRKN5 — Type
FineRKN5()

A 5th order explicit Runge-Kutta-Nyström method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

References

@article{fine1987low,
-  title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods},
-  author={Fine, Jerry Michael},
-  journal={Computing},
-  volume={38},
-  number={4},
-  pages={281--297},
-  year={1987},
-  publisher={Springer}
-}
source
OrdinaryDiffEqRKN.DPRKN6 — Type
DPRKN6

6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative. Free 6th order interpolant.

References

@article{dormand1987runge, title={Runge-kutta-nystrom triples}, author={Dormand, JR and Prince, PJ}, journal={Computers \& Mathematics with Applications}, volume={13}, number={12}, pages={937–949}, year={1987}, publisher={Elsevier} }

source
OrdinaryDiffEqRKN.DPRKN6FM — Type
DPRKN6FM

6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.

Compared to DPRKN6, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250} }

source
OrdinaryDiffEqRKN.DPRKN8 — Type
DPRKN8

8th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.

Not as efficient as DPRKN12 when high accuracy is needed, however this solver is competitive with DPRKN6 at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press} }

source
OrdinaryDiffEqRKN.DPRKN12 — Type
DPRKN12

12th order explicit Rugne-Kutta-Nyström method. The second order ODE should not depend on the first derivative.

Most efficient when high accuracy is needed.

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press} }

source
OrdinaryDiffEqRKN.ERKN4 — Type
ERKN4

Embedded 4(3) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.

The second order ODE should not depend on the first derivative.

Uses adaptive step size control. This method is extra efficient on periodic problems.

References

@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{"o}m Method for Solving Periodic Initial Value Problems}, author={Demba, MA and Senu, N and Ismail, F}, journal={Applied Mathematical Sciences}, volume={11}, number={17}, pages={819–838}, year={2017} }

source
OrdinaryDiffEqRKN.ERKN5 — Type
ERKN5

Embedded 5(4) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.

The second order ODE should not depend on the first derivative.

Uses adaptive step size control. This method is extra efficient on periodic problems.

References

@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{"o}m Methods for the Numerical Solution of Oscillatory Initial Value Problems}, author={Demba, Musa A and Senu, Norazak and Ismail, Fudziah}, journal={Mathematical and Computational Applications}, volume={21}, number={4}, pages={46}, year={2016}, publisher={Multidisciplinary Digital Publishing Institute} }

source
OrdinaryDiffEqRKN.ERKN7 — Type
ERKN7

Embedded pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.

The second order ODE should not depend on the first derivative.

Uses adaptive step size control. This method is extra efficient on periodic Problems.

References

@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{"o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753} }

source
diff --git a/dev/dynamical/symplectic/index.html b/dev/dynamical/symplectic/index.html deleted file mode 100644 index 6cd4339a6b..0000000000 --- a/dev/dynamical/symplectic/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Symplectic Runge-Kutta Methods · OrdinaryDiffEq.jl
GitHub

Symplectic Runge-Kutta Methods

Missing docstring.

Missing docstring for SymplecticEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for VelocityVerlet. Check Documenter's build log for details.

Missing docstring.

Missing docstring for VerletLeapfrog. Check Documenter's build log for details.

Missing docstring.

Missing docstring for PseudoVerletLeapfrog. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Ruth3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CandyRoz4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CalvoSanz4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte42. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Yoshida6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for KahanLi6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for McAte8. Check Documenter's build log for details.

Missing docstring.

Missing docstring for KahanLi8. Check Documenter's build log for details.

OrdinaryDiffEqSymplecticRK.SofSpa10 — Type

@article{sofroniou2005derivation, title={Derivation of symmetric composition constants for symmetric integrators}, author={Sofroniou, Mark and Spaletta, Giulia}, journal={Optimization Methods and Software}, volume={20}, number={4-5}, pages={597–613}, year={2005}, publisher={Taylor \& Francis} }

source
diff --git a/dev/explicit/AdamsBashforthMoulton/index.html b/dev/explicit/AdamsBashforthMoulton/index.html new file mode 100644 index 0000000000..11c23282d3 --- /dev/null +++ b/dev/explicit/AdamsBashforthMoulton/index.html @@ -0,0 +1,17 @@ + +OrdinaryDiffEqAdamsBashforthMoulton · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqAdamsBashforthMoulton

Multistep methods, useful for integrating a very expensive to evaluate non-stiff system of differential equations.

Installation

To be able to access the solvers in OrdinaryDiffEqAdamsBashforthMoulton, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqAdamsBashforthMoulton")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqAdamsBashforthMoulton
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, VCABM())

Full list of solvers

Explicit Multistep Methods

OrdinaryDiffEqAdamsBashforthMoulton.AB3 — Type
AB3()

Adams-Bashforth Explicit Method The 3-step third order multistep method. Ralston's Second Order Method is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.AB4 — Type
AB4()

Adams-Bashforth Explicit Method The 4-step fourth order multistep method. Runge-Kutta method of order 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.AB5 — Type
AB5()

Adams-Bashforth Explicit Method The 5-step fifth order multistep method. Ralston's 3rd order Runge-Kutta method is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source

Predictor-Corrector Methods

OrdinaryDiffEqAdamsBashforthMoulton.ABM32 — Type
ABM32()

Adams-Bashforth Explicit Method It is third order method. In ABM32, AB3 works as predictor and Adams Moulton 2-steps method works as Corrector. Ralston's Second Order Method is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.ABM43 — Type
ABM43()

Adams-Bashforth Explicit Method It is fourth order method. In ABM43, AB4 works as predictor and Adams Moulton 3-steps method works as Corrector. Runge-Kutta method of order 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.ABM54 — Type
ABM54()

Adams-Bashforth Explicit Method It is fifth order method. In ABM54, AB5 works as predictor and Adams Moulton 4-steps method works as Corrector. Runge-Kutta method of order 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCAB3 — Type
VCAB3()

Adams explicit Method The 3rd order Adams method. Bogacki-Shampine 3/2 method is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCAB4 — Type
VCAB4()

Adams explicit Method The 4th order Adams method. Runge-Kutta 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCAB5 — Type
VCAB5()

Adams explicit Method The 5th order Adams method. Runge-Kutta 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCABM3 — Type
VCABM3()

Adams explicit Method The 3rd order Adams-Moulton method. Bogacki-Shampine 3/2 method is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCABM4 — Type
VCABM4()

Adams explicit Method The 4th order Adams-Moulton method. Runge-Kutta 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCABM5 — Type
VCABM5()

Adams explicit Method The 5th order Adams-Moulton method. Runge-Kutta 4 is used to calculate starting values.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
OrdinaryDiffEqAdamsBashforthMoulton.VCABM — Type
VCABM()

adaptive order Adams explicit Method An adaptive order adaptive time Adams Moulton method. It uses an order adaptivity algorithm is derived from Shampine's DDEABM.

Keyword Arguments

References

E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1

source
diff --git a/dev/explicit/Extrapolation/index.html b/dev/explicit/Extrapolation/index.html new file mode 100644 index 0000000000..a22aa1b3c3 --- /dev/null +++ b/dev/explicit/Extrapolation/index.html @@ -0,0 +1,30 @@ + +OrdinaryDiffEqExtrapolation · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqExtrapolation

Solvers based on within method parallelism, allowing multithreading of the solution across different values of f. The explicit extrapolation solvers are generally outclassed by other explicit methods. However, some stiff extrapolation methods perform very well if the problem is sufficiently stiff.

Installation

To be able to access the solvers in OrdinaryDiffEqExtrapolation, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqExtrapolation")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqExtrapolation
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, ExtrapolationMidpointDeuflhard())

Full list of solvers

OrdinaryDiffEqExtrapolation.AitkenNeville — Type
AitkenNeville(; max_order::Int = 10,
+                min_order::Int = 1,
+                init_order = 3,
+                thread = OrdinaryDiffEq.False())

Parallelized Explicit Extrapolation Method. Euler extrapolation using Aitken-Neville with the Romberg Sequence.

Keyword Arguments

  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

source
OrdinaryDiffEqExtrapolation.ExtrapolationMidpointDeuflhard — Type
ExtrapolationMidpointDeuflhard(; max_order = 10,
+                                 min_order = 1,
+                                 init_order = 5,
+                                 thread = OrdinaryDiffEq.True(),
+                                 sequence = :harmonic,
+                                 sequence_factor = 2)

Parallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates.

Keyword Arguments

  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.
  • sequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

source
OrdinaryDiffEqExtrapolation.ExtrapolationMidpointHairerWanner — Type
ExtrapolationMidpointHairerWanner(; max_order = 10,
+                                    min_order = 2,
+                                    init_order = 5,
+                                    thread = OrdinaryDiffEq.True(),
+                                    sequence = :harmonic,
+                                    sequence_factor = 2)

Parallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates, following Hairer's ODEX in the adaptivity behavior.

Keyword Arguments

  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.
  • sequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

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diff --git a/dev/explicit/Feagin/index.html b/dev/explicit/Feagin/index.html new file mode 100644 index 0000000000..39cb9a0cf6 --- /dev/null +++ b/dev/explicit/Feagin/index.html @@ -0,0 +1,17 @@ + +OrdinaryDiffEqFeagin · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqFeagin

Preferred solvers for non-stiff problems at very low tolerance, <1e-30. Best combined with preciser than Float64 number types for the state, such as the BigFloat number type. Note that the Feagin methods have a less robust error estimator than the Verner methods, and thus even for very low tolerance problems the Verner methods (Vern9) may still be more efficient. In addition, at extremely low tolerances the explicit extrapolation methods allow for arbitrarily high variable order stepping which will also outperform the Feagin methods. As such, the Feagin methods may be useful in the Float128 precision range but should be tested against other algorithms.

Installation

To be able to access the solvers in OrdinaryDiffEqFeagin, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqFeagin")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqFeagin
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, Feagin14())

Full list of solvers

OrdinaryDiffEqFeagin.Feagin10 — Type
Feagin10(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Feagin's 10th-order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \& Scientific Computations} }

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OrdinaryDiffEqFeagin.Feagin12 — Type
Feagin12(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Feagin's 12th-order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \& Scientific Computations} }

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OrdinaryDiffEqFeagin.Feagin14 — Type
Feagin14(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Feagin's 14th-order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{feagin2009explicit, title={An Explicit Runge-Kutta Method of Order Fourteen}, author={Feagin, Terry}, year={2009}, publisher={Numerical Algorithms} }

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diff --git a/dev/explicit/HighOrderRK/index.html b/dev/explicit/HighOrderRK/index.html new file mode 100644 index 0000000000..8e4376fa82 --- /dev/null +++ b/dev/explicit/HighOrderRK/index.html @@ -0,0 +1,26 @@ + +OrdinaryDiffEqHighOrderRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqHighOrderRK

Solvers for non-stiff problems at low tolerance. However, the solvers in OrdinaryDiffEqVerner generally perform better at low tolerances.

Installation

To be able to access the solvers in OrdinaryDiffEqHighOrderRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqHighOrderRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqHighOrderRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, DP8())

Full list of solvers

OrdinaryDiffEqHighOrderRK.TanYam7 — Type
TanYam7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Tanaka-Yamashita 7 Runge-Kutta method. (7th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Tanaka M., Muramatsu S., Yamashita S., (1992), On the Optimization of Some Nine-Stage Seventh-order Runge-Kutta Method, Information Processing Society of Japan, 33 (12), pp. 1512-1526.

source
OrdinaryDiffEqHighOrderRK.TsitPap8 — Type
TsitPap8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Tsitouras-Papakostas 8/7 Runge-Kutta method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{tsitouras1999cheap, title={Cheap error estimation for Runge–Kutta methods}, author={Tsitouras, Ch and Papakostas, SN}, journal={SIAM Journal on Scientific Computing}, volume={20}, number={6}, pages={2067–2088}, year={1999}, publisher={SIAM}}

source
OrdinaryDiffEqHighOrderRK.DP8 — Type
DP8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Hairer's 8/5/3 adaption of the Dormand-Prince Runge-Kutta method. (7th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqHighOrderRK.PFRK87 — Type
PFRK87(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         thread = OrdinaryDiffEq.False(),
+         omega = 0.0)

Explicit Runge-Kutta Method. Phase-fitted Runge-Kutta of 8th order.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • omega: a periodicity phase estimate, when accurate this method results in zero numerical dissipation.

References

@article{tsitouras2017phase, title={Phase-fitted Runge–Kutta pairs of orders 8 (7)}, author={Tsitouras, Ch and Famelis, I Th and Simos, TE}, journal={Journal of Computational and Applied Mathematics}, volume={321}, pages={226–231}, year={2017}, publisher={Elsevier}}

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diff --git a/dev/explicit/LowOrderRK/index.html b/dev/explicit/LowOrderRK/index.html new file mode 100644 index 0000000000..9a96e82468 --- /dev/null +++ b/dev/explicit/LowOrderRK/index.html @@ -0,0 +1,68 @@ + +OrdinaryDiffEqLowOrderRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqLowOrderRK

If OrdinaryDiffEqTsit5 is not working well for your non-stiff problem at default and higher tolerance, it can be worthwhile to explore the options in this package. In particular, when more robust error control is required, BS5 is a good choice. If at moderate tolerances and the interpolation error is very important, consider the OwrenZen5 method. For fast solving at higher tolerances, we recommend BS3, or OwrenZen3if the interpolation error is important.

Installation

To be able to access the solvers in OrdinaryDiffEqLowOrderRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqLowOrderRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqLowOrderRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, BS3())

Full list of solvers

OrdinaryDiffEqLowOrderRK.Euler — Type
Euler()

Explicit Runge-Kutta Method. The canonical forward Euler method. Fixed timestep only.

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqLowOrderRK.Heun — Type
Heun(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+       step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+       thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. The second order Heun's method. Uses embedded Euler method for adaptivity.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqLowOrderRK.Ralston — Type
Ralston(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. The optimized second order midpoint method. Uses embedded Euler method for adaptivity.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqLowOrderRK.Midpoint — Type
Midpoint(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. The second order midpoint method. Uses embedded Euler method for adaptivity.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqLowOrderRK.RK4 — Type
RK4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. The canonical Runge-Kutta Order 4 method. Uses a defect control for adaptive stepping using maximum error over the whole interval.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{shampine2005solving, title={Solving ODEs and DDEs with residual control}, author={Shampine, LF}, journal={Applied Numerical Mathematics}, volume={52}, number={1}, pages={113–127}, year={2005}, publisher={Elsevier} }

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OrdinaryDiffEqLowOrderRK.BS3 — Type
BS3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, four-stage FSAL method with embedded error estimator of Bogacki and Shampine.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{bogacki19893, title={A 3 (2) pair of Runge-Kutta formulas}, author={Bogacki, Przemyslaw and Shampine, Lawrence F}, journal={Applied Mathematics Letters}, volume={2}, number={4}, pages={321–325}, year={1989}, publisher={Elsevier} }

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OrdinaryDiffEqLowOrderRK.OwrenZen3 — Type
OwrenZen3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 3/2 method (free 3rd order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }

source
OrdinaryDiffEqLowOrderRK.OwrenZen4 — Type
OwrenZen4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 4/3 method (free 4th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }

source
OrdinaryDiffEqLowOrderRK.OwrenZen5 — Type
OwrenZen5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 5/4 method (free 5th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }

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OrdinaryDiffEqLowOrderRK.BS5 — Type
BS5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False(),
+      lazy = true)

Explicit Runge-Kutta Method. Bogacki-Shampine 5/4 Runge-Kutta method. (lazy 5th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • lazy: determines if the lazy interpolant is used.

References

@article{bogacki1996efficient, title={An efficient runge-kutta (4, 5) pair}, author={Bogacki, P and Shampine, Lawrence F}, journal={Computers \& Mathematics with Applications}, volume={32}, number={6}, pages={15–28}, year={1996}, publisher={Elsevier} }

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OrdinaryDiffEqLowOrderRK.DP5 — Type
DP5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Dormand-Prince's 5/4 Runge-Kutta method. (free 4th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{dormand1980family, title={A family of embedded Runge-Kutta formulae}, author={Dormand, John R and Prince, Peter J}, journal={Journal of computational and applied mathematics}, volume={6}, number={1}, pages={19–26}, year={1980}, publisher={Elsevier} }

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OrdinaryDiffEqLowOrderRK.Anas5 — Type
Anas5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        w = 1)

Explicit Runge-Kutta Method. 4th order Runge-Kutta method designed for periodic problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • w: a periodicity estimate, which when accurate the method becomes 5th order

(and is otherwise 4th order with less error for better estimates).

References

@article{anastassi2005optimized, title={An optimized Runge–Kutta method for the solution of orbital problems}, author={Anastassi, ZA and Simos, TE}, journal={Journal of Computational and Applied Mathematics}, volume={175}, number={1}, pages={1–9}, year={2005}, publisher={Elsevier}}

source
OrdinaryDiffEqLowOrderRK.RKO65 — Type
RKO65(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 5th order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Tsitouras, Ch. "Explicit Runge–Kutta methods for starting integration of Lane–Emden problem." Applied Mathematics and Computation 354 (2019): 353-364. doi: https://doi.org/10.1016/j.amc.2019.02.047

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OrdinaryDiffEqLowOrderRK.FRK65 — Type
FRK65(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        omega = 0.0)

Explicit Runge-Kutta Method. Zero Dissipation Runge-Kutta of 6th order.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • omega: a periodicity phase estimate,

when accurate this method results in zero numerical dissipation.

References

@article{medvedev2018fitted, title={Fitted modifications of Runge-Kutta pairs of orders 6 (5)}, author={Medvedev, Maxim A and Simos, TE and Tsitouras, Ch}, journal={Mathematical Methods in the Applied Sciences}, volume={41}, number={16}, pages={6184–6194}, year={2018}, publisher={Wiley Online Library}}

source
OrdinaryDiffEqLowOrderRK.RKM — Type
RKM(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+      thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Method designed to have good stability properties when applied to pseudospectral discretizations of hyperbolic partial differential equaitons.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{mead1999optimal, title={Optimal Runge–Kutta methods for first order pseudospectral operators}, author={Mead, JL and Renaut, RA}, journal={Journal of Computational Physics}, volume={152}, number={1}, pages={404–419}, year={1999}, publisher={Elsevier} }

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OrdinaryDiffEqLowOrderRK.MSRK5 — Type
MSRK5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 5th order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Misha Stepanov - https://arxiv.org/pdf/2202.08443.pdf : Figure 3.

source
OrdinaryDiffEqLowOrderRK.MSRK6 — Type
MSRK6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 6th order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Misha Stepanov - https://arxiv.org/pdf/2202.08443.pdf : Table4

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OrdinaryDiffEqLowOrderRK.PSRK4p7q6 — Type
PSRK4p7q6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 6-stage Pseudo-Symplectic method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, volume = {38}, PAGES = {439-461}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, volume = {328}, pages = {86-94}, year = {2017}, issn = {0021-9991}, doi = {https://doi.org/10.1016/j.jcp.2016.10.040}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}

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OrdinaryDiffEqLowOrderRK.PSRK3p5q4 — Type
PSRK3p5q4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 4-stage Pseudo-Symplectic method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, year = {2017}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}

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OrdinaryDiffEqLowOrderRK.PSRK3p6q5 — Type
PSRK3p6q5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 5-stage Pseudo-Symplectic method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, year = {2017}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}

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OrdinaryDiffEqLowOrderRK.Stepanov5 — Type
Stepanov5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 5th order method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Stepanov2021Embedded5, title={Embedded (4, 5) pairs of explicit 7-stage Runge–Kutta methods with FSAL property}, author={Misha Stepanov}, journal={Calcolo}, year={2021}, volume={59} }

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OrdinaryDiffEqLowOrderRK.SIR54 — Type
SIR54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 5th order method suited for SIR-type epidemic models.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Kovalnogov2020RungeKuttaPS, title={Runge–Kutta pairs suited for SIR‐type epidemic models}, author={Vladislav N. Kovalnogov and Theodore E. Simos and Ch. Tsitouras}, journal={Mathematical Methods in the Applied Sciences}, year={2020}, volume={44}, pages={5210 - 5216} }

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OrdinaryDiffEqLowOrderRK.Alshina2 — Type
Alshina2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 2nd order, 2-stage Method with optimal parameters.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }

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OrdinaryDiffEqLowOrderRK.Alshina3 — Type
Alshina3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 3rd order, 3-stage Method with optimal parameters.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }

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OrdinaryDiffEqLowOrderRK.Alshina6 — Type
Alshina6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 6th order, 7-stage Method with optimal parameters.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }

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diff --git a/dev/explicit/LowStorageRK/index.html b/dev/explicit/LowStorageRK/index.html new file mode 100644 index 0000000000..8b21eee2fa --- /dev/null +++ b/dev/explicit/LowStorageRK/index.html @@ -0,0 +1,117 @@ + +OrdinaryDiffEqLowStorageRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqLowStorageRK

These methods are designed to have reduced register requirements, allowing for larger sets of ODEs to more easily fit into RAM. For example, while the 5th order Tsit5 requires around 9 concurrent instantiations of the ODE state u, RDPK3Sp510 can achieve 5th order with 3 registers, effectively requiring 1/3 of the memory. However, there are some efficiency trade-offs used in the design of the low-storage RK methods, and thus they are generally only recommended in situations which are RAM bound, like large-scale PDE discretizations.

Installation

To be able to access the solvers in OrdinaryDiffEqLowStorageRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqLowStorageRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqLowStorageRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, CarpenterKennedy2N54())

Full list of solvers

OrdinaryDiffEqLowStorageRK.ORK256 — Type
ORK256(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         thread = OrdinaryDiffEq.False(),
+         williamson_condition = true)

Explicit Runge-Kutta Method. A second-order, five-stage method for wave propagation equations. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Matteo Bernardini, Sergio Pirozzoli. A General Strategy for the Optimization of Runge-Kutta Schemes for Wave Propagation Phenomena. Journal of Computational Physics, 228(11), pp 4182-4199, 2009. doi: https://doi.org/10.1016/j.jcp.2009.02.032

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OrdinaryDiffEqLowStorageRK.DGLDDRK73_C — Type
DGLDDRK73_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False(),
+              williamson_condition = true)

Explicit Runge-Kutta Method. 7-stage, third order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

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OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54 — Type
CarpenterKennedy2N54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                       step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                       thread = OrdinaryDiffEq.False(),
+                       williamson_condition = true)

Explicit Runge-Kutta Method. A fourth-order, five-stage low-storage method of Carpenter and Kennedy (free 3rd order Hermite interpolant). Fixed timestep only. Designed for hyperbolic PDEs (stability properties).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

@article{carpenter1994fourth, title={Fourth-order 2N-storage Runge-Kutta schemes}, author={Carpenter, Mark H and Kennedy, Christopher A}, year={1994} }

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OrdinaryDiffEqLowStorageRK.NDBLSRK124 — Type
NDBLSRK124(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False(),
+             williamson_condition = true)

Explicit Runge-Kutta Method. 12-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

source
OrdinaryDiffEqLowStorageRK.NDBLSRK144 — Type
NDBLSRK144(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False(),
+             williamson_condition = true)

Explicit Runge-Kutta Method. 14-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

source
OrdinaryDiffEqLowStorageRK.CFRLDDRK64 — Type
CFRLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-storage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

M. Calvo, J. M. Franco, L. Randez. A New Minimum Storage Runge–Kutta Scheme for Computational Acoustics. Journal of Computational Physics, 201, pp 1-12, 2004. doi: https://doi.org/10.1016/j.jcp.2004.05.012

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OrdinaryDiffEqLowStorageRK.TSLDDRK74 — Type
TSLDDRK74(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 7-stage, fourth order low-storage low-dissipation, low-dispersion scheme with maximal accuracy and stability limit along the imaginary axes. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Kostas Tselios, T. E. Simos. Optimized Runge–Kutta Methods with Minimal Dispersion and Dissipation for Problems arising from Computational Acoustics. Physics Letters A, 393(1-2), pp 38-47, 2007. doi: https://doi.org/10.1016/j.physleta.2006.10.072

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OrdinaryDiffEqLowStorageRK.DGLDDRK84_C — Type
DGLDDRK84_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False(),
+              williamson_condition = true)

Explicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

source
OrdinaryDiffEqLowStorageRK.DGLDDRK84_F — Type
DGLDDRK84_F(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False(),
+              williamson_condition = true)

Explicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when the maximum spatial step size is not constrained. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

source
OrdinaryDiffEqLowStorageRK.SHLDDRK64 — Type
SHLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False(),
+            williamson_condition = true)

Explicit Runge-Kutta Method. A fourth-order, six-stage low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

D. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }

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OrdinaryDiffEqLowStorageRK.RK46NL — Type
RK46NL(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Julien Berland, Christophe Bogey, Christophe Bailly. Low-Dissipation and Low-Dispersion Fourth-Order Runge-Kutta Algorithm. Computers & Fluids, 35(10), pp 1459-1463, 2006. doi: https://doi.org/10.1016/j.compfluid.2005.04.003

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OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32 — Type
ParsaniKetchesonDeconinck3S32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 3-stage, second order (3S) low-storage scheme, optimized the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

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OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82 — Type
ParsaniKetchesonDeconinck3S82(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, second order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53 — Type
ParsaniKetchesonDeconinck3S53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173 — Type
ParsaniKetchesonDeconinck3S173(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 17-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94 — Type
ParsaniKetchesonDeconinck3S94(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184 — Type
ParsaniKetchesonDeconinck3S184(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 18-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105 — Type
ParsaniKetchesonDeconinck3S105(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 10-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205 — Type
ParsaniKetchesonDeconinck3S205(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 20-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.CKLLSRK43_2 — Type
CKLLSRK43_2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 4-stage, third order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C — Type
CKLLSRK54_3C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S — Type
CKLLSRK95_4S(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C — Type
CKLLSRK95_4C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M — Type
CKLLSRK95_4M(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R — Type
CKLLSRK54_3C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R — Type
CKLLSRK54_3M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R — Type
CKLLSRK54_3N_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R — Type
CKLLSRK85_4C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R — Type
CKLLSRK85_4M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R — Type
CKLLSRK85_4P_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R — Type
CKLLSRK54_3N_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R — Type
CKLLSRK54_3M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R — Type
CKLLSRK65_4M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. 6-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R — Type
CKLLSRK85_4FM_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                   step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                   thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R — Type
CKLLSRK75_4M_5R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. CKLLSRK754M5R: Low-Storage Method 7-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp35 — Type
RDPK3Sp35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35 — Type
RDPK3SpFSAL35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp49 — Type
RDPK3Sp49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fourth-order, nine-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49 — Type
RDPK3SpFSAL49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fourth-order, nine-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp510 — Type
RDPK3Sp510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fifth-order, ten-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510 — Type
RDPK3SpFSAL510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                 step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                 thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fifth-order, ten-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.HSLDDRK64 — Type
HSLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False(),
+            williamson_condition = true)

Explicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

D. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }

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OrdinaryDiffEqLowStorageRK.NDBLSRK134 — Type
NDBLSRK134(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False(),
+             williamson_condition = true)

Explicit Runge-Kutta Method. 13-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

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OrdinaryDiffEqLowStorageRK.SHLDDRK_2N — Type
SHLDDRK_2N(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low dissipation and dispersion Runge-Kutta schemes for computational acoustics

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{stanescu19982n, title={2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics}, author={Stanescu, D and Habashi, WG}, journal={Journal of Computational Physics}, volume={143}, number={2}, pages={674–681}, year={1998}, publisher={Elsevier}}

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OrdinaryDiffEqLowStorageRK.SHLDDRK52 — Type
SHLDDRK52(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Low dissipation and dispersion Runge-Kutta schemes for computational acoustics

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{stanescu19982n, title={2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics}, author={Stanescu, D and Habashi, WG}, journal={Journal of Computational Physics}, volume={143}, number={2}, pages={674–681}, year={1998}, publisher={Elsevier}}

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diff --git a/dev/explicit/Nordsieck/index.html b/dev/explicit/Nordsieck/index.html new file mode 100644 index 0000000000..b4ba0de969 --- /dev/null +++ b/dev/explicit/Nordsieck/index.html @@ -0,0 +1,17 @@ + +OrdinaryDiffEqNordsieck · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqNordsieck

The Nordsieck form is an alternative representation of multistep methods which, instead of representing and saving past step values in a history vector, it uses a derivative list (like a Taylor expansion) for the computation of the next point. The Nordsieck form was pioneered by early implementations of BDF methods such LSODE, VODE, and finally CVODE. It can have some advantages in terms of restartability as the full Nordsieck vector can be instantiated given only the information of f and its derivatives after discontinuities, but the higher derivative representations can also introduce numerical instabilities of their own.

The Nordsieck implementations here are considered experimental implementations of the LSODE non-fixed leading coefficient form and are generally considered inferior to the fixed-leading history-based BDF implementation of FBDF, and thus for all standard usage we recommend FBDF. However, this algorithm is kept for experimental research and development purposes with the possibility of one day becoming a more discontinuity-aware BDF implementation.

Installation

To be able to access the solvers in OrdinaryDiffEqNordsieck, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqNordsieck")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqNordsieck
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, AN5())

Full list of solvers

OrdinaryDiffEqNordsieck.AN5 — Type
AN5()

Adaptive step size Adams explicit Method An adaptive 5th order fixed-leading coefficient Adams method in Nordsieck form.

Experimental

AN5 is experimental, the solver VCABM is generally preferred.

Keyword Arguments

References

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diff --git a/dev/explicit/PRK/index.html b/dev/explicit/PRK/index.html new file mode 100644 index 0000000000..4de814018f --- /dev/null +++ b/dev/explicit/PRK/index.html @@ -0,0 +1,17 @@ + +OrdinaryDiffEqPRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqPRK

Explicit solvers optimized for a certain number of parallel calls of the system of ordinary differential equations f.

Installation

To be able to access the solvers in OrdinaryDiffEqPRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqPRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqPRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, KuttaPRK2p5())

Full list of solvers

OrdinaryDiffEqPRK.KuttaPRK2p5 — Type
KuttaPRK2p5(; thread = OrdinaryDiffEq.True())

Explicit Runge-Kutta Method A 5 parallel, 2 processor method of 5th order.

Keyword Arguments

  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{jackson1995potential, title={The potential for parallelism in Runge–Kutta methods. Part 1: RK formulas in standard form}, author={Jackson, Kenneth R and Norsett, Syvert Paul}, journal={SIAM journal on numerical analysis}, volume={32}, number={1}, pages={49–82}, year={1995}, publisher={SIAM}}

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diff --git a/dev/explicit/QPRK/index.html b/dev/explicit/QPRK/index.html new file mode 100644 index 0000000000..35639537ad --- /dev/null +++ b/dev/explicit/QPRK/index.html @@ -0,0 +1,19 @@ + +OrdinaryDiffEqQPRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqQPRK

Explicit solvers optimized for a certain number of parallel calls of the system of ordinary differential equations f. Particularly good at low tolerances, when using quad-precision arithmetic, Float128.

Installation

To be able to access the solvers in OrdinaryDiffEqQPRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqQPRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqQPRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, QPRK98())

Full list of solvers

OrdinaryDiffEqQPRK.QPRK98 — Type
QPRK98(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Runge–Kutta pairs of orders 9(8) for use in quadruple precision computations

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Kovalnogov VN, Fedorov RV, Karpukhina TV, Simos TE, Tsitouras C. Runge–Kutta pairs of orders 9 (8) for use in quadruple precision computations. Numerical Algorithms, 2023. doi: https://doi.org/10.1007/s11075-023-01632-8

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diff --git a/dev/explicit/RKN/index.html b/dev/explicit/RKN/index.html new file mode 100644 index 0000000000..b2120e2760 --- /dev/null +++ b/dev/explicit/RKN/index.html @@ -0,0 +1,18 @@ + +OrdinaryDiffEqRKN · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqRKN

Second order solvers.

To be able to access the solvers in OrdinaryDiffEqRKN, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqRKN")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqOrdinaryDiffEqRKN
+function HH_acceleration!(dv, v, u, p, t)
+    x, y = u
+    dx, dy = dv
+    dv[1] = -x - 2x * y
+    dv[2] = y^2 - y - x^2
+end
+initial_positions = [0.0, 0.1]
+initial_velocities = [0.5, 0.0]
+tspan = (0.0, 1.0)
+prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
+sol = solve(prob, Nystrom4(), dt = 1 / 10)

Full list of solvers

OrdinaryDiffEqRKN.IRKN3 — Type
IRKN3()

Improved Runge-Kutta-Nyström method Method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}

source
OrdinaryDiffEqRKN.IRKN4 — Type
IRKN4()

Improved Runge-Kutta-Nyström method Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not be dependent on the first derivative. Recommended for smooth problems with expensive functions to evaluate.

Keyword Arguments

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}

source
OrdinaryDiffEqRKN.Nystrom4 — Type
Nystrom4()

Improved Runge-Kutta-Nyström method A 4th order explicit method which can be applied directly on second order ODEs. Can only be used with fixed time steps. In case the ODE Problem is not dependent on the first derivative consider using Nystrom4VelocityIndependent to increase performance.

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.Nystrom4VelocityIndependent — Type
Nystrom4VelocityIndependent()

Improved Runge-Kutta-Nyström method A 4th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.Nystrom5VelocityIndependent — Type
Nystrom5VelocityIndependent()

Improved Runge-Kutta-Nyström method A 5th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

source
OrdinaryDiffEqRKN.FineRKN4 — Type
FineRKN4()

Improved Runge-Kutta-Nyström method A 4th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

Keyword Arguments

References

@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}

source
OrdinaryDiffEqRKN.FineRKN5 — Type
FineRKN5()

Improved Runge-Kutta-Nyström method A 5th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

Keyword Arguments

References

@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}

source
OrdinaryDiffEqRKN.DPRKN4 — Type
DPRKN4()

Improved Runge-Kutta-Nyström method 4th order explicit method. The second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

source
OrdinaryDiffEqRKN.DPRKN5 — Type
DPRKN5()

Improved Runge-Kutta-Nyström method 5th order explicit method. The second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{Bettis1973ARN, title={A Runge-Kutta Nystrom algorithm}, author={Dale G. Bettis}, journal={Celestial mechanics}, year={1973}, volume={8}, pages={229-233}, publisher={Springer}}

source
OrdinaryDiffEqRKN.DPRKN6 — Type
DPRKN6()

Improved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Free 6th order interpolant

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

source
OrdinaryDiffEqRKN.DPRKN6FM — Type
DPRKN6FM()

Improved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Compared to DPRKN6, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

source
OrdinaryDiffEqRKN.DPRKN8 — Type
DPRKN8()

Improved Runge-Kutta-Nyström method 8th order explicit method. The second order ODE should not depend on the first derivative. Not as efficient as DPRKN12 when high accuracy is needed, however this solver is competitive with DPRKN6 at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.

Keyword Arguments

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}

source
OrdinaryDiffEqRKN.DPRKN12 — Type
DPRKN12()

Improved Runge-Kutta-Nyström method 12th order explicit method. The second order ODE should not depend on the first derivative. Most efficient when high accuracy is needed.

Keyword Arguments

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}

source
OrdinaryDiffEqRKN.ERKN4 — Type
ERKN4()

Improved Runge-Kutta-Nyström method Embedded 4(3) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{"o}m Method for Solving Periodic Initial Value Problems}, author={Demba, MA and Senu, N and Ismail, F}, journal={Applied Mathematical Sciences}, volume={11}, number={17}, pages={819–838}, year={2017}}

source
OrdinaryDiffEqRKN.ERKN5 — Type
ERKN5()

Improved Runge-Kutta-Nyström method Embedded 5(4) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{"o}m Methods for the Numerical Solution of Oscillatory Initial Value Problems}, author={Demba, Musa A and Senu, Norazak and Ismail, Fudziah}, journal={Mathematical and Computational Applications}, volume={21}, number={4}, pages={46}, year={2016}, publisher={Multidisciplinary Digital Publishing Institute}}

source
OrdinaryDiffEqRKN.ERKN7 — Type
ERKN7()

Improved Runge-Kutta-Nyström method Embedded pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{"o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753}}

source
OrdinaryDiffEqRKN.RKN4 — Type
RKN4()

Improved Runge-Kutta-Nyström method 3 stage fourth order method to solve second order linear inhomogeneous IVPs. Does not include an adaptive method. Solves for for d-dimensional differential systems of second order linear inhomogeneous equations.

Warning

This method is only fourth order for these systems, the method is second order otherwise!

Keyword Arguments

References

@article{MONTIJANO2024115533, title = {Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs}, author = {J.I. Montijano and L. Rández and M. Calvo}, journal = {Journal of Computational and Applied Mathematics}, volume = {438}, pages = {115533}, year = {2024},}

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diff --git a/dev/explicit/SSPRK/index.html b/dev/explicit/SSPRK/index.html new file mode 100644 index 0000000000..007083a873 --- /dev/null +++ b/dev/explicit/SSPRK/index.html @@ -0,0 +1,53 @@ + +OrdinaryDiffEqSSPRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqSSPRK

SSPRK methods are Runge-Kutta methods which support the "strongly preserving property" (SSP). They are designed for the use in discretizations of hyperbolic partial differential equations and conservation laws as they have extra stability properties ( e.g., stability with respect to total variation, the maximum norm, or other convex functionals) when step-size restrictions are respected. In particular, these properties are granted if the step-size is kept to a level where the CFL coefficients are less than the SSP coefficient.

Note that for SSPRK methods, a algorithm utility OrdinaryDiffEqCore.ssp_coefficient(alg) exists that allows for querying the SSP coefficient for use in step size calculations.

Installation

To be able to access the solvers in OrdinaryDiffEqSSPRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqSSPRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqSSPRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, SSPRK22())

Full list of solvers

OrdinaryDiffEqSSPRK.SSPRK22 — Type
SSPRK22(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A second-order, two-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Shu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5

source
OrdinaryDiffEqSSPRK.SSPRK33 — Type
SSPRK33(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, three-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Shu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5

source
OrdinaryDiffEqSSPRK.SSPRK53 — Type
SSPRK53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.KYKSSPRK42 — Type
KYKSSPRK42(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kubatko2014optimal, title={Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods}, author={Kubatko, Ethan J and Yeager, Benjamin A and Ketcheson, David I}, journal={Journal of Scientific Computing}, volume={60}, pages={313–344}, year={2014}, publisher={Springer}}

source
OrdinaryDiffEqSSPRK.KYK2014DGSSPRK_3S2 — Type
KYK2014DGSSPRK_3S2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                     step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                     thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{kubatko2014optimal, title={Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods}, author={Kubatko, Ethan J and Yeager, Benjamin A and Ketcheson, David I}, journal={Journal of Scientific Computing}, volume={60}, pages={313–344}, year={2014}, publisher={Springer}}

source
OrdinaryDiffEqSSPRK.SSPRK53_2N1 — Type
SSPRK53_2N1(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK53_2N2 — Type
SSPRK53_2N2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK53_H — Type
SSPRK53_H(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK63 — Type
SSPRK63(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, six-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK73 — Type
SSPRK73(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, seven-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK83 — Type
SSPRK83(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, eight-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK43 — Type
SSPRK43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Optimal third-order explicit SSP method with four stages discovered by

  • J. F. B. M. Kraaijevanger. "Contractivity of Runge-Kutta methods." In: BIT Numerical Mathematics 31.3 (1991), pp. 482–528. DOI: 10.1007/BF01933264.

Embedded method constructed by

  • Sidafa Conde, Imre Fekete, John N. Shadid. "Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods." arXiv: 1806.08693

Efficient implementation (and optimized controller) developed by

  • Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836
source
OrdinaryDiffEqSSPRK.SSPRK432 — Type
SSPRK432(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011. Example 6.1

source
OrdinaryDiffEqSSPRK.SSPRKMSVS43 — Type
SSPRKMSVS43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, four-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Shu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073

source
OrdinaryDiffEqSSPRK.SSPRKMSVS32 — Type
SSPRKMSVS32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A second-order, three-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Shu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073

source
OrdinaryDiffEqSSPRK.SSPRK932 — Type
SSPRK932(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A third-order, nine-stage explicit strong stability preserving (SSP) method.

Consider using SSPRK43 instead, which uses the same main method and an improved embedded method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011.

source
OrdinaryDiffEqSSPRK.SSPRK54 — Type
SSPRK54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fourth-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207.

source
OrdinaryDiffEqSSPRK.SSPRK104 — Type
SSPRK104(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fourth-order, ten-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

Ketcheson, David I. Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM Journal on Scientific Computing 30.4 (2008): 2113-2136.

source
diff --git a/dev/explicit/SymplecticRK/index.html b/dev/explicit/SymplecticRK/index.html new file mode 100644 index 0000000000..b7aef3e321 --- /dev/null +++ b/dev/explicit/SymplecticRK/index.html @@ -0,0 +1,18 @@ + +OrdinaryDiffEqSymplecticRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqSymplecticRK

A symplectic integrator is an integrator whose solution resides on a symplectic manifold. Because of discretization error, when it is solving a Hamiltonian system it doesn't get exactly the correct trajectory on the manifold. Instead, that trajectory itself is perturbed O(Δtn) for the order n from the true trajectory. Then there's a linear drift due to numerical error of this trajectory over time Normal integrators tend to have a quadratic (or more) drift, and do not have any good global guarantees about this phase space path (just local). What means is that symplectic integrators tend to capture the long-time patterns better than normal integrators because of this lack of drift and this almost guarantee of periodicity.

Installation

To be able to access the solvers in OrdinaryDiffEqSymplecticRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqSymplecticRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqSymplecticRK
+function HH_acceleration!(dv, v, u, p, t)
+    x, y = u
+    dx, dy = dv
+    dv[1] = -x - 2x * y
+    dv[2] = y^2 - y - x^2
+end
+initial_positions = [0.0, 0.1]
+initial_velocities = [0.5, 0.0]
+tspan = (0.0, 1.0)
+prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
+sol = solve(prob, KahanLi8(), dt = 1 / 10)

Full list of solvers

OrdinaryDiffEqSymplecticRK.VelocityVerlet — Type
VelocityVerlet()

Symplectic Runge-Kutta Methods 2nd order explicit symplectic integrator. Requires f_2(t,u) = v, i.e. a second order ODE.

Keyword Arguments

References

@article{verlet1967computer, title={Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }

source
OrdinaryDiffEqSymplecticRK.VerletLeapfrog — Type
VerletLeapfrog()

Symplectic Runge-Kutta Methods 2nd order explicit symplectic integrator.

Keyword Arguments

References

@article{verlet1967computer, title={Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }

source
OrdinaryDiffEqSymplecticRK.PseudoVerletLeapfrog — Type
PseudoVerletLeapfrog()

Symplectic Runge-Kutta Methods 2nd order explicit symplectic integrator.

Keyword Arguments

References

@article{verlet1967computer, title={Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }

source
OrdinaryDiffEqSymplecticRK.McAte2 — Type
McAte2()

Symplectic Runge-Kutta Methods Optimized efficiency 2nd order explicit symplectic integrator.

Keyword Arguments

References

@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }

source
OrdinaryDiffEqSymplecticRK.Ruth3 — Type
Ruth3()

Symplectic Runge-Kutta Methods 3rd order explicit symplectic integrator.

Keyword Arguments

References

@article{ruth1983canonical, title={A canonical integration technique}, author={Ruth, Ronald D}, journal={IEEE Trans. Nucl. Sci.}, volume={30}, number={CERN-LEP-TH-83-14}, pages={2669–2671}, year={1983}}

source
OrdinaryDiffEqSymplecticRK.McAte3 — Type
McAte3()

Symplectic Runge-Kutta Methods Optimized efficiency 3rd order explicit symplectic integrator.

Keyword Arguments

References

@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }

source
OrdinaryDiffEqSymplecticRK.CandyRoz4 — Type
CandyRoz4()

Symplectic Runge-Kutta Methods 4th order explicit symplectic integrator.

Keyword Arguments

References

@article{candy1991symplectic, itle={A symplectic integration algorithm for separable Hamiltonian functions}, uthor={Candy, J and Rozmus, W}, ournal={Journal of Computational Physics}, olume={92}, umber={1}, ages={230–256}, ear={1991}, ublisher={Elsevier}}

source
OrdinaryDiffEqSymplecticRK.McAte4 — Type
McAte4()

Symplectic Runge-Kutta Methods 4th order explicit symplectic integrator. Requires quadratic kinetic energy.

Keyword Arguments

References

@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }

source
OrdinaryDiffEqSymplecticRK.CalvoSanz4 — Type
CalvoSanz4()

Symplectic Runge-Kutta Methods Optimized efficiency 4th order explicit symplectic integrator.

Keyword Arguments

References

@article{sanz1993symplectic, title={Symplectic numerical methods for Hamiltonian problems}, author={Sanz-Serna, Jes{'u}s Maria and Calvo, Mari-Paz}, journal={International Journal of Modern Physics C}, volume={4}, number={02}, pages={385–392}, year={1993}, publisher={World Scientific} }

source
OrdinaryDiffEqSymplecticRK.McAte42 — Type
McAte42()

Symplectic Runge-Kutta Methods 4th order explicit symplectic integrator. BROKEN

Keyword Arguments

References

@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }

source
OrdinaryDiffEqSymplecticRK.McAte5 — Type
McAte5()

Symplectic Runge-Kutta Methods Optimized efficiency 5th order explicit symplectic integrator. Requires quadratic kinetic energy.

Keyword Arguments

References

@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }

source
OrdinaryDiffEqSymplecticRK.Yoshida6 — Type
Yoshida6()

Symplectic Runge-Kutta Methods 6th order explicit symplectic integrator.

Keyword Arguments

References

@article{yoshida1990construction, title={Construction of higher order symplectic integrators}, author={Yoshida, Haruo}, journal={Physics letters A}, volume={150}, number={5-7}, pages={262–268}, year={1990}, publisher={Elsevier}}

source
OrdinaryDiffEqSymplecticRK.KahanLi6 — Type
KahanLi6()

Symplectic Runge-Kutta Methods Optimized efficiency 6th order explicit symplectic integrator.

Keyword Arguments

References

@article{yoshida1990construction, title={Construction of higher order symplectic integrators}, author={Yoshida, Haruo}, journal={Physics letters A}, volume={150}, number={5-7}, pages={262–268}, year={1990}, publisher={Elsevier}}

source
OrdinaryDiffEqSymplecticRK.McAte8 — Type
McAte8()

Symplectic Runge-Kutta Methods 8th order explicit symplectic integrator.

Keyword Arguments

References

@article{mclachlan1995numerical, title={On the numerical integration of ordinary differential equations by symmetric composition methods}, author={McLachlan, Robert I}, journal={SIAM Journal on Scientific Computing}, volume={16}, number={1}, pages={151–168}, year={1995}, publisher={SIAM} }

source
OrdinaryDiffEqSymplecticRK.KahanLi8 — Type
KahanLi8()

Symplectic Runge-Kutta Methods Optimized efficiency 8th order explicit symplectic integrator.

Keyword Arguments

References

@article{kahan1997composition, title={Composition constants for raising the orders of unconventional schemes for ordinary differential equations}, author={Kahan, William and Li, Ren-Cang}, journal={Mathematics of computation}, volume={66}, number={219}, pages={1089–1099}, year={1997}}

source
OrdinaryDiffEqSymplecticRK.SofSpa10 — Type
SofSpa10()

Symplectic Runge-Kutta Methods 10th order explicit symplectic integrator.

Keyword Arguments

References

@article{sofroniou2005derivation, title={Derivation of symmetric composition constants for symmetric integrators}, author={Sofroniou, Mark and Spaletta, Giulia}, journal={Optimization Methods and Software}, volume={20}, number={4-5}, pages={597–613}, year={2005}, publisher={Taylor \& Francis}}

source
diff --git a/dev/explicit/Tsit5/index.html b/dev/explicit/Tsit5/index.html new file mode 100644 index 0000000000..7963e3611c --- /dev/null +++ b/dev/explicit/Tsit5/index.html @@ -0,0 +1,19 @@ + +OrdinaryDiffEqTsit5 · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqTsit5

Recommended solver for most non-stiff problems at default and higher tolerance.

Installation

To be able to access the solvers in OrdinaryDiffEqTsit5, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqTsit5")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqTsit5
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, Tsit5())

Full list of solvers

OrdinaryDiffEqTsit5.Tsit5 — Type
Tsit5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False())

Explicit Runge-Kutta Method. A fifth-order explicit Runge-Kutta method with embedded error estimator of Tsitouras. Free 4th order interpolant.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

@article{tsitouras2011runge, title={Runge–Kutta pairs of order 5 (4) satisfying only the first column simplifying assumption}, author={Tsitouras, Ch}, journal={Computers \& Mathematics with Applications}, volume={62}, number={2}, pages={770–775}, year={2011}, publisher={Elsevier} }

source
OrdinaryDiffEqTsit5.AutoTsit5 — Function

Automatic switching algorithm that can switch between the (non-stiff) Tsit5() and stiff_alg.

AutoTsit5(stiff_alg; kwargs...)

This method is equivalent to AutoAlgSwitch(Tsit5(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.

source
diff --git a/dev/explicit/Verner/index.html b/dev/explicit/Verner/index.html new file mode 100644 index 0000000000..351e700ba0 --- /dev/null +++ b/dev/explicit/Verner/index.html @@ -0,0 +1,29 @@ + +OrdinaryDiffEqVerner · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqVerner

Preferred solvers for non-stiff problems at low tolerance. Vern6, Vern7, or Vern8 are good methods for tolerances between ~1e-8-1e-12, and using Float64 numbers for the state of the differential equation. For even lower tolerances,Vern9 should be used, combined with the more precise BigFloat number type.

Installation

To be able to access the solvers in OrdinaryDiffEqVerner, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqVerner")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqVerner
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, Vern6())

Full list of solvers

OrdinaryDiffEqVerner.Vern6 — Type
Vern6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 6/5 Runge-Kutta method. (lazy 6th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

source
OrdinaryDiffEqVerner.Vern7 — Type
Vern7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 7/6 Runge-Kutta method. (lazy 7th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

source
OrdinaryDiffEqVerner.Vern8 — Type
Vern8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 8/7 Runge-Kutta method. (lazy 8th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

source
OrdinaryDiffEqVerner.Vern9 — Type
Vern9(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        thread = OrdinaryDiffEq.False(),
+        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 9/8 Runge-Kutta method. (lazy9th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

source
OrdinaryDiffEqVerner.AutoVern6 — Function

Automatic switching algorithm that can switch between the (non-stiff) Vern6() and stiff_alg.

AutoVern6(stiff_alg; kwargs...)

This method is equivalent to AutoAlgSwitch(Vern6(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.

source
OrdinaryDiffEqVerner.AutoVern7 — Function

Automatic switching algorithm that can switch between the (non-stiff) Vern7() and stiff_alg.

AutoVern7(stiff_alg; kwargs...)

This method is equivalent to AutoAlgSwitch(Vern7(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.

source
OrdinaryDiffEqVerner.AutoVern8 — Function

Automatic switching algorithm that can switch between the (non-stiff) Vern8() and stiff_alg.

AutoVern8(stiff_alg; kwargs...)

This method is equivalent to AutoAlgSwitch(Vern8(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.

source
OrdinaryDiffEqVerner.AutoVern9 — Function

Automatic switching algorithm that can switch between the (non-stiff) Vern9() and stiff_alg.

AutoVern9(stiff_alg; kwargs...)

This method is equivalent to AutoAlgSwitch(Vern9(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.

source
diff --git a/dev/imex/IMEXMultistep/index.html b/dev/imex/IMEXMultistep/index.html new file mode 100644 index 0000000000..343e9f7650 --- /dev/null +++ b/dev/imex/IMEXMultistep/index.html @@ -0,0 +1,13 @@ + +OrdinaryDiffEqIMEXMultistep · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqIMEXMultistep

Solvers if your system of ordinary differential equations can be split up into the sum of a stiff and non-stiff part.

Installation

To be able to access the solvers in OrdinaryDiffEqIMEXMultistep, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqIMEXMultistep")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqIMEXMultistep
+f1 = (u, p, t) -> 2u
+f2 = (u, p, t) -> 2u
+u0 = 1.0
+tspan = (0.0, 1.0)
+prob = SplitODEProblem(f1, f2, u0, tspan)
+sol = solve(prob,CNAB2(), dt = 1 / 10)

Full list of solvers

OrdinaryDiffEqIMEXMultistep.CNAB2 — Type
CNAB2()

IMEX Multistep method. Crank-Nicholson Adams-Bashforth 2.

Keyword Arguments

References

@article{jorgenson2014unconditional, title={Unconditional stability of a Crank-Nicolson Adams-Bashforth 2 numerical method}, author={JORGENSON, ANDREW D}, journal={A (A- C)}, volume={1}, number={2}, pages={1}, year={2014}} @article{he2010numerical, title={Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time-dependent Navier–Stokes equations}, author={He, Yinnian and Li, Jian}, journal={International journal for numerical methods in fluids}, volume={62}, number={6}, pages={647–659}, year={2010}, publisher={Wiley Online Library}}

source
OrdinaryDiffEqIMEXMultistep.CNLF2 — Type
CNLF2()

IMEX Multistep method. Crank-Nicholson Leapfrong 2.

Keyword Arguments

References

@article{han2020second, title={A second order, linear, unconditionally stable, Crank–Nicolson–Leapfrog scheme for phase field models of two-phase incompressible flows}, author={Han, Daozhi and Jiang, Nan}, journal={Applied Mathematics Letters}, volume={108}, pages={106521}, year={2020}, publisher={Elsevier}} @article{jiang2015crank, title={A Crank–Nicolson Leapfrog stabilization: Unconditional stability and two applications}, author={Jiang, Nan and Kubacki, Michaela and Layton, William and Moraiti, Marina and Tran, Hoang}, journal={Journal of Computational and Applied Mathematics}, volume={281}, pages={263–276}, year={2015}, publisher={Elsevier}}

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diff --git a/dev/imex/imex_multistep/index.html b/dev/imex/imex_multistep/index.html deleted file mode 100644 index 9d11cc5be6..0000000000 --- a/dev/imex/imex_multistep/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -IMEX Multistep Methods · OrdinaryDiffEq.jl
GitHub

IMEX Multistep Methods

Missing docstring.

Missing docstring for CNAB2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CNLF2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SBDF. Check Documenter's build log for details.

OrdinaryDiffEqBDF.SBDF2 — Function
SBDF2(;kwargs...)

The two-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

source
OrdinaryDiffEqBDF.SBDF3 — Function
SBDF3(;kwargs...)

The three-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

source
OrdinaryDiffEqBDF.SBDF4 — Function
SBDF4(;kwargs...)

The four-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

source
diff --git a/dev/imex/imex_sdirk/index.html b/dev/imex/imex_sdirk/index.html deleted file mode 100644 index f5fa3bed84..0000000000 --- a/dev/imex/imex_sdirk/index.html +++ /dev/null @@ -1,7 +0,0 @@ - -IMEX SDIRK Methods · OrdinaryDiffEq.jl
GitHub

IMEX SDIRK Methods

OrdinaryDiffEqBDF.IMEXEuler — Function
IMEXEuler(;kwargs...)

The one-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

When applied to a SplitODEProblem of the form

u'(t) = f1(u) + f2(u)

The default IMEXEuler() method uses an update of the form

unew = uold + dt * (f1(unew) + f2(uold))

See also SBDF, IMEXEulerARK.

source
OrdinaryDiffEqBDF.IMEXEulerARK — Function
IMEXEulerARK(;kwargs...)

The one-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

When applied to a SplitODEProblem of the form

u'(t) = f1(u) + f2(u)

A classical additive Runge-Kutta method in the sense of Araújo, Murua, Sanz-Serna (1997) consisting of the implicit and the explicit Euler method given by

y1   = uold + dt * f1(y1)
-unew = uold + dt * (f1(unew) + f2(y1))

See also SBDF, IMEXEuler.

source
OrdinaryDiffEqSDIRK.KenCarp3 — Type

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }

KenCarp3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting

source
OrdinaryDiffEqSDIRK.KenCarp4 — Type

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }

KenCarp4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method with splitting

source
OrdinaryDiffEqSDIRK.KenCarp47 — Type

@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier} }

KenCarp47: SDIRK Method An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting

source
OrdinaryDiffEqSDIRK.KenCarp5 — Type

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }

KenCarp5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method with splitting

source
OrdinaryDiffEqSDIRK.KenCarp58 — Type

@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier} }

KenCarp58: SDIRK Method An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting

source
Missing docstring.

Missing docstring for ESDIRK54I8L2SA. Check Documenter's build log for details.

OrdinaryDiffEqSDIRK.ESDIRK436L2SA2 — Type

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK437L2SA — Type

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK547L2SA2 — Type

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK659L2SA — Type

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244}

Currently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details. }

source
diff --git a/dev/implicit/BDF/index.html b/dev/implicit/BDF/index.html new file mode 100644 index 0000000000..25c039f0b9 --- /dev/null +++ b/dev/implicit/BDF/index.html @@ -0,0 +1,117 @@ + +OrdinaryDiffEqBDF · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqBDF

Multistep methods, good for large stiff systems.

Installation

To be able to access the solvers in OrdinaryDiffEqBDF, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqBDF")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqBDF
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, QNDF())

Full list of solvers

OrdinaryDiffEqBDF.ABDF2 — Type
ABDF2(; chunk_size = Val{0}(),
+        autodiff = true,
+        standardtag = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:forward},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS,
+        κ = nothing,
+        tol = nothing,
+        nlsolve = NLNewton(),
+        smooth_est = true,
+        extrapolant = :linear,
+        controller = :Standard,
+        step_limiter! = trivial_limiter!)

Multistep Method. An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ABDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- κ: TBD
  • tol: TBD
  • nlsolve: TBD
  • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

E. Alberdi Celayaa, J. J. Anza Aguirrezabalab, P. Chatzipantelidisc. Implementation of an Adaptive BDF2 Formula and Comparison with The MATLAB Ode15s. Procedia Computer Science, 29, pp 1014-1026, 2014. doi: https://doi.org/10.1016/j.procs.2014.05.091

source
OrdinaryDiffEqBDF.QNDF — Type
QNDF(; chunk_size = Val{0}(),
+       autodiff = true,
+       standardtag = Val{true}(),
+       concrete_jac = nothing,
+       diff_type = Val{:forward},
+       linsolve = nothing,
+       precs = DEFAULT_PRECS,
+       κ = nothing,
+       tol = nothing,
+       nlsolve = NLNewton(),
+       extrapolant = :linear,
+       kappa =  promote(-0.1850, -1 // 9, -0.0823, -0.0415, 0),
+       controller = :Standard,
+       step_limiter! = trivial_limiter!)

Multistep Method. An adaptive order quasi-constant timestep NDF method. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- κ: TBD
  • tol: TBD
  • nlsolve: TBD
  • extrapolant: TBD
  • kappa: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }

source
OrdinaryDiffEqBDF.QNDF1 — Type
QNDF1(; chunk_size = Val{0}(),
+        autodiff = true,
+        standardtag = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:forward},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS,
+        nlsolve = NLNewton(),
+        extrapolant = :linear,
+        kappa = -0.1850,
+        controller = :Standard,
+        step_limiter! = trivial_limiter!)

Multistep Method. An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine's accuracy-optimal -0.1850.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF1(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- nlsolve: TBD
  • extrapolant: TBD
  • kappa: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }

source
OrdinaryDiffEqBDF.QNDF2 — Type
QNDF2(; chunk_size = Val{0}(),
+        autodiff = true,
+        standardtag = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:forward},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS,
+        nlsolve = NLNewton(),
+        extrapolant = :linear,
+        kappa =  -1 // 9,
+        controller = :Standard,
+        step_limiter! = trivial_limiter!)

Multistep Method. An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- nlsolve: TBD
  • extrapolant: TBD
  • kappa: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }

source
OrdinaryDiffEqBDF.MEBDF2 — Type
MEBDF2(; chunk_size = Val{0}(),
+         autodiff = true,
+         standardtag = Val{true}(),
+         concrete_jac = nothing,
+         diff_type = Val{:forward},
+         linsolve = nothing,
+         precs = DEFAULT_PRECS,
+         nlsolve = NLNewton(),
+         extrapolant = :constant)

Multistep Method. The second order Modified Extended BDF method, which has improved stability properties over the standard BDF. Fixed timestep only.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify MEBDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- nlsolve: TBD
  • extrapolant: TBD

References

@article{cash2000modified, title={Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs}, author={Cash, JR}, journal={Journal of Computational and Applied Mathematics}, volume={125}, number={1-2}, pages={117–130}, year={2000}, publisher={Elsevier}}

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OrdinaryDiffEqBDF.FBDF — Type
FBDF(; chunk_size = Val{0}(),
+       autodiff = true,
+       standardtag = Val{true}(),
+       concrete_jac = nothing,
+       diff_type = Val{:forward},
+       linsolve = nothing,
+       precs = DEFAULT_PRECS,
+       κ = nothing,
+       tol = nothing,
+       nlsolve = NLNewton(),
+       extrapolant = :linear,
+       controller = :Standard,
+       step_limiter! = trivial_limiter!,
+       max_order::Val{MO} = Val{5}())

Multistep Method. An adaptive order quasi-constant timestep NDF method. Fixed leading coefficient BDF. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify FBDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- κ: TBD
  • tol: TBD
  • nlsolve: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • max_order: TBD

References

@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH \& Co. KG}}

source

IMEX Multistep

Missing docstring.

Missing docstring for SBDF. Check Documenter's build log for details.

OrdinaryDiffEqBDF.SBDF2 — Function
SBDF2(;kwargs...)

The two-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

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OrdinaryDiffEqBDF.SBDF3 — Function
SBDF3(;kwargs...)

The three-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

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OrdinaryDiffEqBDF.SBDF4 — Function
SBDF4(;kwargs...)

The four-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

See also SBDF.

source

IMEX SDIRK

OrdinaryDiffEqBDF.IMEXEuler — Function
IMEXEuler(;kwargs...)

The one-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

When applied to a SplitODEProblem of the form

u'(t) = f1(u) + f2(u)

The default IMEXEuler() method uses an update of the form

unew = uold + dt * (f1(unew) + f2(uold))

See also SBDF, IMEXEulerARK.

source
OrdinaryDiffEqBDF.IMEXEulerARK — Function
IMEXEulerARK(;kwargs...)

The one-step version of the IMEX multistep methods of

  • Uri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037

When applied to a SplitODEProblem of the form

u'(t) = f1(u) + f2(u)

A classical additive Runge-Kutta method in the sense of Araújo, Murua, Sanz-Serna (1997) consisting of the implicit and the explicit Euler method given by

y1   = uold + dt * f1(y1)
+unew = uold + dt * (f1(unew) + f2(y1))

See also SBDF, IMEXEuler.

source

DAE

OrdinaryDiffEqBDF.DImplicitEuler — Type
DImplicitEuler(; chunk_size = Val{0}(),
+                 autodiff = true,
+                 standardtag = Val{true}(),
+                 concrete_jac = nothing,
+                 diff_type = Val{:forward},
+                 linsolve = nothing,
+                 precs = DEFAULT_PRECS,
+                 nlsolve = NLNewton(),
+                 extrapolant = :constant,
+                 controller = :Standard)

Multistep Method. Implicit Euler for implicit DAE form. It uses an apriori error estimator for adaptivity based on a finite differencing approximation from SPICE.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DImplicitEuler(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- nlsolve: TBD
  • extrapolant: TBD
  • controller: TBD

References

source
OrdinaryDiffEqBDF.DABDF2 — Type
DABDF2(; chunk_size = Val{0}(),
+         autodiff = true,
+         standardtag = Val{true}(),
+         concrete_jac = nothing,
+         diff_type = Val{:forward},
+         linsolve = nothing,
+         precs = DEFAULT_PRECS,
+         nlsolve = NLNewton(),
+         extrapolant = :constant,
+         controller = :Standard)

Multistep Method. Fully implicit implementation of BDF2.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DABDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- nlsolve: TBD
  • extrapolant: TBD
  • controller: TBD

References

@article{celaya2014implementation, title={Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s}, author={Celaya, E Alberdi and Aguirrezabala, JJ Anza and Chatzipantelidis, Panagiotis}, journal={Procedia Computer Science}, volume={29}, pages={1014–1026}, year={2014}, publisher={Elsevier}}

source
OrdinaryDiffEqBDF.DFBDF — Type
DFBDF(; chunk_size = Val{0}(),
+        autodiff = true,
+        standardtag = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:forward},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS,
+        κ = nothing,
+        tol = nothing,
+        nlsolve = NLNewton(),
+        extrapolant = :linear,
+        controller = :Standard,
+        max_order::Val{MO} = Val{5}())

Multistep Method. Fully implicit implementation of FBDF based on Shampine's

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DFBDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
    /n- κ: TBD
  • tol: TBD
  • nlsolve: TBD
  • extrapolant: TBD
  • controller: TBD
  • max_order: TBD

References

@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH and Co. KG} }

source
diff --git a/dev/implicit/Extrapolation/index.html b/dev/implicit/Extrapolation/index.html new file mode 100644 index 0000000000..f134374345 --- /dev/null +++ b/dev/implicit/Extrapolation/index.html @@ -0,0 +1,62 @@ + +OrdinaryDiffEqExtrapolation · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqExtrapolation

Solvers based on within method parallelism. These solvers perform well for medium sized systems of ordinary differential equations, of about 20 to 500 equations, at low tolerances.

Installation

To be able to access the solvers in OrdinaryDiffEqExtrapolation, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqExtrapolation")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqExtrapolation
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, ImplicitEulerBarycentricExtrapolation())

Full list of solvers

OrdinaryDiffEqExtrapolation.ImplicitEulerExtrapolation — Type
ImplicitEulerExtrapolation(; chunk_size = Val{0}(),
+                             autodiff = true,
+                             standardtag = Val{true}(),
+                             concrete_jac = nothing,
+                             diff_type = Val{:forward},
+                             linsolve = nothing,
+                             precs = DEFAULT_PRECS,
+                             max_order = 12,
+                             min_order = 3,
+                             init_order = 5,
+                             thread = OrdinaryDiffEq.False(),
+                             sequence = :harmonic)

Parallelized Explicit Extrapolation Method. Extrapolation of implicit Euler method with Romberg sequence. Similar to Hairer's SEULEX.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEulerExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

source
OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation — Type
ImplicitDeuflhardExtrapolation(; chunk_size = Val{0}(),
+                                 autodiff = true,
+                                 standardtag = Val{true}(),
+                                 concrete_jac = nothing,
+                                 diff_type = Val{:forward},
+                                 linsolve = nothing,
+                                 precs = DEFAULT_PRECS,
+                                 max_order = 10,
+                                 min_order = 1,
+                                 init_order = 5,
+                                 thread = OrdinaryDiffEq.False(),
+                                 sequence = :harmonic)

Parallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitDeuflhardExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

source
OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation — Type
ImplicitHairerWannerExtrapolation(; chunk_size = Val{0}(),
+                                    autodiff = true,
+                                    standardtag = Val{true}(),
+                                    concrete_jac = nothing,
+                                    diff_type = Val{:forward},
+                                    linsolve = nothing,
+                                    precs = DEFAULT_PRECS,
+                                    max_order = 10,
+                                    min_order = 2,
+                                    init_order = 5,
+                                    thread = OrdinaryDiffEq.False(),
+                                    sequence = :harmonic)

Parallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitHairerWannerExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

source
OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation — Type
ImplicitEulerBarycentricExtrapolation(; chunk_size = Val{0}(),
+                                        autodiff = true,
+                                        standardtag = Val{true}(),
+                                        concrete_jac = nothing,
+                                        diff_type = Val{:forward},
+                                        linsolve = nothing,
+                                        precs = DEFAULT_PRECS,
+                                        max_order = 10,
+                                        min_order = 3,
+                                        init_order = 5,
+                                        thread = OrdinaryDiffEq.False(),
+                                        sequence = :harmonic,
+                                        sequence_factor = 2)

Parallelized Explicit Extrapolation Method. Euler extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEulerBarycentricExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • max_order: maximum order of the adaptive order algorithm.
  • min_order: minimum order of the adaptive order algorithm.
  • init_order: initial order of the adaptive order algorithm.
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.
  • sequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.
  • sequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.

References

@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}

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diff --git a/dev/implicit/FIRK/index.html b/dev/implicit/FIRK/index.html new file mode 100644 index 0000000000..9841c10f95 --- /dev/null +++ b/dev/implicit/FIRK/index.html @@ -0,0 +1,44 @@ + +OrdinaryDiffEqFIRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqFIRK

FIRK methods are fully implicit Runge-Kutta methods. They can have special properties, like be symplectic integrators, and can achieve higher order for the same number of stage in comparison to diagonal methods. However, the fully implicit methods have a larger implicit system to solve and thus have a higher linear algebra cost. This can be useful in some contexts to promote more parallelism, but also since the size of the factorization is cubic and the dominant cost for large equations, multiplying O(n^3) operations to O((sn)^3) can be a considerable cost increase for FIRK tableaus, where s, the number of stages, is particularly large. That said, the restriction to diagonal implicitness imposes order restrictions, such as SDIRK methods having a maximum order of 5, which can restrict the problems best suited for SDIRK methods.

The most common FIRK method in production are those based on RadauIIA tableaus, which is an ODE representation of Gaussian collocation. Like Gaussian collocation, it achieves higher order convergence than its stages, namely order 2s+1 for s stages. Thus RadauIIA FIRK methods tend to be some of the highest order methods (excluding extrapolation methods). This means that high order RadauIIA methods are recommended in the same scenarios that high-order explicit Runge-Kutta methods are recommended simply with the restriction of being a stiff equation. Such scenarios include cases like very low tolerances: RadauIIA methods can be the best performing methods for scenarios where tolerances are 1e-9 and below. Additionally, for ODE systems of size less than 200, the increased size of the Jacobian is mitigated by improved multithreading, since BLAS implementations are only good at multithreading LU factorizations after a certain matrix size. For this reason, RadauIIA methods tend to be recommended in cases where ODE size is small to intermediate and very accurate solutions are required.

They should be tested against the parallel implicit extrapolation which also specialize in this regime.

Installation

To be able to access the solvers in OrdinaryDiffEqFIRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqFIRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqFIRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, RadauIIA5())

Full list of solvers

OrdinaryDiffEqFIRK.RadauIIA3 — Type
RadauIIA3(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            extrapolant = :dense,
+            smooth_est = true,
+            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source
OrdinaryDiffEqFIRK.RadauIIA5 — Type
RadauIIA5(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            extrapolant = :dense,
+            smooth_est = true,
+            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source
OrdinaryDiffEqFIRK.RadauIIA9 — Type
RadauIIA9(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            extrapolant = :dense,
+            smooth_est = true,
+            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA9(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source
diff --git a/dev/implicit/PDIRK/index.html b/dev/implicit/PDIRK/index.html new file mode 100644 index 0000000000..f20361ad3f --- /dev/null +++ b/dev/implicit/PDIRK/index.html @@ -0,0 +1,26 @@ + +OrdinaryDiffEqPDIRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqPDIRK

PDIRK methods are parallel DIRK methods. SDIRK methods, or singly-diagonally implicit methods, have to build and solve a factorize a Jacobian of the form W = I-gammaJ where gamma is dependent on the chosen method. PDIRK methods use multiple different choices of gamma, i.e. W_i = I-gamma_iJ, which are all used in the update process. There are some advantages to this, as no SDIRK method can be a higher order than 5, while DIRK methods generally can have arbitrarily high order and lower error coefficients, leading to lower errors at larger dt sizes. With the right construction of the tableau, these matrices can be factorized and the underlying steps can be computed in parallel, which is why these are the parallel DIRK methods.

Experimental

OrdinaryDiffEqPDIRK is experimental, as there are no parallel DIRK tableaus that achieve good performance in the literature.

Installation

To be able to access the solvers in OrdinaryDiffEqPDIRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqPDIRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqPDIRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, PDIRK44())

Full list of solvers

OrdinaryDiffEqPDIRK.PDIRK44 — Type
PDIRK44(; chunk_size = Val{0}(),
+          autodiff = true,
+          standardtag = Val{true}(),
+          concrete_jac = nothing,
+          diff_type = Val{:forward},
+          linsolve = nothing,
+          precs = DEFAULT_PRECS,
+          nlsolve = NLNewton(),
+          extrapolant = :constant,
+          thread = OrdinaryDiffEq.True())

Parallel Diagonally Implicit Runge-Kutta Method. A 2 processor 4th order diagonally non-adaptive implicit method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify PDIRK44(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD,
  • extrapolant: TBD,
  • thread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.

References

"@article{iserles1990theory, title={On the theory of parallel Runge—Kutta methods}, author={Iserles, Arieh and Norrsett, SP}, journal={IMA Journal of numerical Analysis}, volume={10}, number={4}, pages={463–488}, year={1990}, publisher={Oxford University Press}}

source
diff --git a/dev/implicit/Rosenbrock/index.html b/dev/implicit/Rosenbrock/index.html new file mode 100644 index 0000000000..0975e36612 --- /dev/null +++ b/dev/implicit/Rosenbrock/index.html @@ -0,0 +1,1280 @@ + +OrdinaryDiffEqRosenbrock · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqRosenbrock

Methods for small and medium sized stiff systems of differential equations. At high tolerances, >1e-2, Rosenbrock23 is a good choice. At medium tolerances >1e-8 it is recommended you use Rodas5P or Rodas4P, the former is more efficient, but the latter is more reliable. For larger systems look at multistep methods.

Installation

To be able to access the solvers in OrdinaryDiffEqRosenbrock, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqRosenbrock")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqRosenbrock
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, Rodas5P())

Full list of solvers

OrdinaryDiffEqRosenbrock.Rosenbrock23 — Type
Rosenbrock23(; chunk_size = Val{0}(),
+               standardtag = Val{true}(),
+               autodiff = Val{true}(),
+               concrete_jac = nothing,
+               diff_type = Val{:central},
+               linsolve = nothing,
+               precs = DEFAULT_PRECS,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock-Wanner Method. An Order 2/3 L-Stable Rosenbrock-W method which is good for very stiff equations with oscillations at low tolerances. 2nd order stiff-aware interpolation.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rosenbrock23(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Shampine L.F. and Reichelt M., (1997) The MATLAB ODE Suite, SIAM Journal of

Scientific Computing, 18 (1), pp. 1-22.

source
OrdinaryDiffEqRosenbrock.Rosenbrock32 — Type
Rosenbrock32(; chunk_size = Val{0}(),
+               standardtag = Val{true}(),
+               autodiff = Val{true}(),
+               concrete_jac = nothing,
+               diff_type = Val{:central},
+               linsolve = nothing,
+               precs = DEFAULT_PRECS,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock-Wanner Method. An Order 3/2 A-Stable Rosenbrock-W method which is good for mildly stiff equations without oscillations at low tolerances. Note that this method is prone to instability in the presence of oscillations, so use with caution. 2nd order stiff-aware interpolation.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rosenbrock32(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Shampine L.F. and Reichelt M., (1997) The MATLAB ODE Suite, SIAM Journal of

Scientific Computing, 18 (1), pp. 1-22.

source
OrdinaryDiffEqRosenbrock.ROS3P — Type
ROS3P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+            and thus uses the internal ForwardDiff.jl algorithm for the choice.
+        - `standardtag`: Specifies whether to use package-specific tags instead of the
+            ForwardDiff default function-specific tags. For more information, see
+            [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+            Defaults to `Val{true}()`.
+        - `autodiff`: Specifies whether to use automatic differentiation via
+            [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+            differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+            Defaults to `Val{true}()` for automatic differentiation.
+        - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+            `nothing`, which means it will be chosen true/false depending on circumstances
+            of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+        - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+            Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+            `Val{:complex}`.
+        - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+          For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+          `ROS3P(linsolve = KLUFactorization()`).
+           When `nothing` is passed, uses `DefaultLinearSolver`.
+        - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+          can be used as a left or right preconditioner.
+          Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+          function where the arguments are defined as:
+            - `W`: the current Jacobian of the nonlinear system. Specified as either
+                ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                representation of the operator. Users can construct the W-matrix on demand
+                by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                the `jac_prototype`.
+            - `du`: the current ODE derivative
+            - `u`: the current ODE state
+            - `p`: the ODE parameters
+            - `t`: the current ODE time
+            - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                the last call to `precs`. It is recommended that this is checked to only
+                update the preconditioner when `newW == true`.
+            - `Plprev`: the previous `Pl`.
+            - `Prprev`: the previous `Pr`.
+            - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                Solver-dependent and subject to change.
+          The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+          To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+          which is not used. Additionally, `precs` must supply the dispatch:
+          ```julia
+          Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+          ```
+          which is used in the solver setup phase to construct the integrator
+          type with the preconditioners `(Pl,Pr)`.
+          The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+          is defined as:
+          ```julia
+          DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+          ```
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method. Keeps high accuracy on discretizations of nonlinear parabolic PDEs.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Lang, J. & Verwer, ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems J. BIT Numerical Mathematics (2001) 41: 731. doi:10.1023/A:1021900219772
source
OrdinaryDiffEqRosenbrock.Rodas3 — Type
Rodas3(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `Rodas3(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G. Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package. In: BIT Numerical Mathematics, 63(2), 2023
source
OrdinaryDiffEqRosenbrock.Rodas23W — Type
Rodas23W(; chunk_size = Val{0}(),
+           standardtag = Val{true}(),
+           autodiff = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:central},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock-Wanner Method. An Order 2/3 L-Stable Rosenbrock-W method for stiff ODEs and DAEs in mass matrix form. 2nd order stiff-aware interpolation and additional error test for interpolation.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas23W(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.
source
OrdinaryDiffEqRosenbrock.Rodas3P — Type
Rodas3P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+              and thus uses the internal ForwardDiff.jl algorithm for the choice.
+          - `standardtag`: Specifies whether to use package-specific tags instead of the
+              ForwardDiff default function-specific tags. For more information, see
+              [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+              Defaults to `Val{true}()`.
+          - `autodiff`: Specifies whether to use automatic differentiation via
+              [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+              differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+              Defaults to `Val{true}()` for automatic differentiation.
+          - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+              `nothing`, which means it will be chosen true/false depending on circumstances
+              of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+          - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+              Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+              `Val{:complex}`.
+          - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+            For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+            `Rodas3P(linsolve = KLUFactorization()`).
+             When `nothing` is passed, uses `DefaultLinearSolver`.
+          - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+            can be used as a left or right preconditioner.
+            Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+            function where the arguments are defined as:
+              - `W`: the current Jacobian of the nonlinear system. Specified as either
+                  ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                  commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                  representation of the operator. Users can construct the W-matrix on demand
+                  by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                  the `jac_prototype`.
+              - `du`: the current ODE derivative
+              - `u`: the current ODE state
+              - `p`: the ODE parameters
+              - `t`: the current ODE time
+              - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                  the last call to `precs`. It is recommended that this is checked to only
+                  update the preconditioner when `newW == true`.
+              - `Plprev`: the previous `Pl`.
+              - `Prprev`: the previous `Pr`.
+              - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                  Solver-dependent and subject to change.
+            The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+            To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+            which is not used. Additionally, `precs` must supply the dispatch:
+            ```julia
+            Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+            ```
+            which is used in the solver setup phase to construct the integrator
+            type with the preconditioners `(Pl,Pr)`.
+            The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+            is defined as:
+            ```julia
+            DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+            ```
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant and additional error test for interpolation. Keeps accuracy on discretizations of linear parabolic PDEs.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.
source
OrdinaryDiffEqRosenbrock.Rodas4 — Type
Rodas4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `Rodas4(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 4th order L-stable Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)
source
OrdinaryDiffEqRosenbrock.Rodas42 — Type
Rodas42(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+              and thus uses the internal ForwardDiff.jl algorithm for the choice.
+          - `standardtag`: Specifies whether to use package-specific tags instead of the
+              ForwardDiff default function-specific tags. For more information, see
+              [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+              Defaults to `Val{true}()`.
+          - `autodiff`: Specifies whether to use automatic differentiation via
+              [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+              differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+              Defaults to `Val{true}()` for automatic differentiation.
+          - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+              `nothing`, which means it will be chosen true/false depending on circumstances
+              of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+          - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+              Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+              `Val{:complex}`.
+          - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+            For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+            `Rodas42(linsolve = KLUFactorization()`).
+             When `nothing` is passed, uses `DefaultLinearSolver`.
+          - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+            can be used as a left or right preconditioner.
+            Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+            function where the arguments are defined as:
+              - `W`: the current Jacobian of the nonlinear system. Specified as either
+                  ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                  commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                  representation of the operator. Users can construct the W-matrix on demand
+                  by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                  the `jac_prototype`.
+              - `du`: the current ODE derivative
+              - `u`: the current ODE state
+              - `p`: the ODE parameters
+              - `t`: the current ODE time
+              - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                  the last call to `precs`. It is recommended that this is checked to only
+                  update the preconditioner when `newW == true`.
+              - `Plprev`: the previous `Pl`.
+              - `Prprev`: the previous `Pr`.
+              - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                  Solver-dependent and subject to change.
+            The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+            To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+            which is not used. Additionally, `precs` must supply the dispatch:
+            ```julia
+            Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+            ```
+            which is used in the solver setup phase to construct the integrator
+            type with the preconditioners `(Pl,Pr)`.
+            The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+            is defined as:
+            ```julia
+            DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+            ```
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)
source
OrdinaryDiffEqRosenbrock.Rodas4P — Type
Rodas4P(; chunk_size = Val{0}(),
+          standardtag = Val{true}(),
+          autodiff = Val{true}(),
+          concrete_jac = nothing,
+          diff_type = Val{:central},
+          linsolve = nothing,
+          precs = DEFAULT_PRECS,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock-Wanner Method. 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems (as opposed to lower if not corrected).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas4P(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.
source
OrdinaryDiffEqRosenbrock.Rodas4P2 — Type
Rodas4P2(; chunk_size = Val{0}(),
+           standardtag = Val{true}(),
+           autodiff = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:central},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock-Wanner Method. A 4th order L-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems. It is an improvement of Roadas4P and in case of inexact Jacobians a second order W method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas4P2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.
source
OrdinaryDiffEqRosenbrock.Rodas5 — Type
Rodas5(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `Rodas5(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Di Marzo G. RODAS5(4) – Méthodes de Rosenbrock d’ordre 5(4) adaptées aux problemes différentiels-algébriques. MSc mathematics thesis, Faculty of Science, University of Geneva, Switzerland.
source
OrdinaryDiffEqRosenbrock.Rodas5P — Type
Rodas5P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+              and thus uses the internal ForwardDiff.jl algorithm for the choice.
+          - `standardtag`: Specifies whether to use package-specific tags instead of the
+              ForwardDiff default function-specific tags. For more information, see
+              [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+              Defaults to `Val{true}()`.
+          - `autodiff`: Specifies whether to use automatic differentiation via
+              [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+              differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+              Defaults to `Val{true}()` for automatic differentiation.
+          - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+              `nothing`, which means it will be chosen true/false depending on circumstances
+              of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+          - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+              Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+              `Val{:complex}`.
+          - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+            For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+            `Rodas5P(linsolve = KLUFactorization()`).
+             When `nothing` is passed, uses `DefaultLinearSolver`.
+          - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+            can be used as a left or right preconditioner.
+            Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+            function where the arguments are defined as:
+              - `W`: the current Jacobian of the nonlinear system. Specified as either
+                  ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                  commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                  representation of the operator. Users can construct the W-matrix on demand
+                  by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                  the `jac_prototype`.
+              - `du`: the current ODE derivative
+              - `u`: the current ODE state
+              - `p`: the ODE parameters
+              - `t`: the current ODE time
+              - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                  the last call to `precs`. It is recommended that this is checked to only
+                  update the preconditioner when `newW == true`.
+              - `Plprev`: the previous `Pl`.
+              - `Prprev`: the previous `Pr`.
+              - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                  Solver-dependent and subject to change.
+            The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+            To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+            which is not used. Additionally, `precs` must supply the dispatch:
+            ```julia
+            Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+            ```
+            which is used in the solver setup phase to construct the integrator
+            type with the preconditioners `(Pl,Pr)`.
+            The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+            is defined as:
+            ```julia
+            DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+            ```
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G. Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package. In: BIT Numerical Mathematics, 63(2), 2023
source
OrdinaryDiffEqRosenbrock.Rodas5Pe — Type
Rodas5Pe(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+               and thus uses the internal ForwardDiff.jl algorithm for the choice.
+           - `standardtag`: Specifies whether to use package-specific tags instead of the
+               ForwardDiff default function-specific tags. For more information, see
+               [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+               Defaults to `Val{true}()`.
+           - `autodiff`: Specifies whether to use automatic differentiation via
+               [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+               differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+               Defaults to `Val{true}()` for automatic differentiation.
+           - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+               `nothing`, which means it will be chosen true/false depending on circumstances
+               of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+           - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+               Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+               `Val{:complex}`.
+           - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+             For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+             `Rodas5Pe(linsolve = KLUFactorization()`).
+              When `nothing` is passed, uses `DefaultLinearSolver`.
+           - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+             can be used as a left or right preconditioner.
+             Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+             function where the arguments are defined as:
+               - `W`: the current Jacobian of the nonlinear system. Specified as either
+                   ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                   commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                   representation of the operator. Users can construct the W-matrix on demand
+                   by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                   the `jac_prototype`.
+               - `du`: the current ODE derivative
+               - `u`: the current ODE state
+               - `p`: the ODE parameters
+               - `t`: the current ODE time
+               - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                   the last call to `precs`. It is recommended that this is checked to only
+                   update the preconditioner when `newW == true`.
+               - `Plprev`: the previous `Pl`.
+               - `Prprev`: the previous `Pr`.
+               - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                   Solver-dependent and subject to change.
+             The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+             To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+             which is not used. Additionally, `precs` must supply the dispatch:
+             ```julia
+             Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+             ```
+             which is used in the solver setup phase to construct the integrator
+             type with the preconditioners `(Pl,Pr)`.
+             The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+             is defined as:
+             ```julia
+             DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+             ```
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G. Rosenbrock methods within OrdinaryDiffEq.jl - Overview, recent developments and applications - Preprint 2024 https://github.com/hbrs-cse/RosenbrockMethods/blob/main/paper/JuliaPaper.pdf
source
OrdinaryDiffEqRosenbrock.Rodas5Pr — Type
Rodas5Pr(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+               and thus uses the internal ForwardDiff.jl algorithm for the choice.
+           - `standardtag`: Specifies whether to use package-specific tags instead of the
+               ForwardDiff default function-specific tags. For more information, see
+               [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+               Defaults to `Val{true}()`.
+           - `autodiff`: Specifies whether to use automatic differentiation via
+               [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+               differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+               Defaults to `Val{true}()` for automatic differentiation.
+           - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+               `nothing`, which means it will be chosen true/false depending on circumstances
+               of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+           - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+               Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+               `Val{:complex}`.
+           - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+             For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+             `Rodas5Pr(linsolve = KLUFactorization()`).
+              When `nothing` is passed, uses `DefaultLinearSolver`.
+           - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+             can be used as a left or right preconditioner.
+             Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+             function where the arguments are defined as:
+               - `W`: the current Jacobian of the nonlinear system. Specified as either
+                   ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                   commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                   representation of the operator. Users can construct the W-matrix on demand
+                   by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                   the `jac_prototype`.
+               - `du`: the current ODE derivative
+               - `u`: the current ODE state
+               - `p`: the ODE parameters
+               - `t`: the current ODE time
+               - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                   the last call to `precs`. It is recommended that this is checked to only
+                   update the preconditioner when `newW == true`.
+               - `Plprev`: the previous `Pl`.
+               - `Prprev`: the previous `Pr`.
+               - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                   Solver-dependent and subject to change.
+             The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+             To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+             which is not used. Additionally, `precs` must supply the dispatch:
+             ```julia
+             Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+             ```
+             which is used in the solver setup phase to construct the integrator
+             type with the preconditioners `(Pl,Pr)`.
+             The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+             is defined as:
+             ```julia
+             DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+             ```
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Steinebach G. Rosenbrock methods within OrdinaryDiffEq.jl - Overview, recent developments and applications - Preprint 2024 https://github.com/hbrs-cse/RosenbrockMethods/blob/main/paper/JuliaPaper.pdf
source
OrdinaryDiffEqRosenbrock.RosenbrockW6S4OS — Type
RosenbrockW6S4OS(; chunk_size = Val{0}(),
+                   standardtag = Val{true}(),
+                   autodiff = Val{true}(),
+                   concrete_jac = nothing,
+                   diff_type = Val{:central},
+                   linsolve = nothing,
+                   precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method (fixed step only).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RosenbrockW6S4OS(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

https://doi.org/10.1016/j.cam.2009.09.017

source
OrdinaryDiffEqRosenbrock.ROS2 — Type
ROS2(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+           and thus uses the internal ForwardDiff.jl algorithm for the choice.
+       - `standardtag`: Specifies whether to use package-specific tags instead of the
+           ForwardDiff default function-specific tags. For more information, see
+           [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+           Defaults to `Val{true}()`.
+       - `autodiff`: Specifies whether to use automatic differentiation via
+           [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+           differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+           Defaults to `Val{true}()` for automatic differentiation.
+       - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+           `nothing`, which means it will be chosen true/false depending on circumstances
+           of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+       - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+           Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+           `Val{:complex}`.
+       - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+         For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+         `ROS2(linsolve = KLUFactorization()`).
+          When `nothing` is passed, uses `DefaultLinearSolver`.
+       - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+         can be used as a left or right preconditioner.
+         Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+         function where the arguments are defined as:
+           - `W`: the current Jacobian of the nonlinear system. Specified as either
+               ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+               commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+               representation of the operator. Users can construct the W-matrix on demand
+               by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+               the `jac_prototype`.
+           - `du`: the current ODE derivative
+           - `u`: the current ODE state
+           - `p`: the ODE parameters
+           - `t`: the current ODE time
+           - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+               the last call to `precs`. It is recommended that this is checked to only
+               update the preconditioner when `newW == true`.
+           - `Plprev`: the previous `Pl`.
+           - `Prprev`: the previous `Pr`.
+           - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+               Solver-dependent and subject to change.
+         The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+         To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+         which is not used. Additionally, `precs` must supply the dispatch:
+         ```julia
+         Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+         ```
+         which is used in the solver setup phase to construct the integrator
+         type with the preconditioners `(Pl,Pr)`.
+         The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+         is defined as:
+         ```julia
+         DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+         ```)

Rosenbrock Method. A 2nd order L-stable Rosenbrock method with 2 internal stages.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • J. G. Verwer et al. (1999): A second-order Rosenbrock method applied to photochemical dispersion problems https://doi.org/10.1137/S1064827597326651
source
OrdinaryDiffEqRosenbrock.ROS2PR — Type
ROS2PR(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `ROS2PR(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```)

Rosenbrock Method. 2nd order stiffly accurate Rosenbrock method with 3 internal stages with (Rinf=0). For problems with medium stiffness the convergence behaviour is very poor and it is recommended to use ROS2S instead.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.ROS2S — Type
ROS2S(; chunk_size = Val{0}(),
+        standardtag = Val{true}(),
+        autodiff = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:central},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. 2nd order stiffly accurate Rosenbrock-Wanner W-method with 3 internal stages with B_PR consistent of order 2 with (Rinf=0).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS2S(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.ROS3 — Type
ROS3(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+           and thus uses the internal ForwardDiff.jl algorithm for the choice.
+       - `standardtag`: Specifies whether to use package-specific tags instead of the
+           ForwardDiff default function-specific tags. For more information, see
+           [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+           Defaults to `Val{true}()`.
+       - `autodiff`: Specifies whether to use automatic differentiation via
+           [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+           differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+           Defaults to `Val{true}()` for automatic differentiation.
+       - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+           `nothing`, which means it will be chosen true/false depending on circumstances
+           of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+       - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+           Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+           `Val{:complex}`.
+       - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+         For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+         `ROS3(linsolve = KLUFactorization()`).
+          When `nothing` is passed, uses `DefaultLinearSolver`.
+       - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+         can be used as a left or right preconditioner.
+         Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+         function where the arguments are defined as:
+           - `W`: the current Jacobian of the nonlinear system. Specified as either
+               ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+               commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+               representation of the operator. Users can construct the W-matrix on demand
+               by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+               the `jac_prototype`.
+           - `du`: the current ODE derivative
+           - `u`: the current ODE state
+           - `p`: the ODE parameters
+           - `t`: the current ODE time
+           - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+               the last call to `precs`. It is recommended that this is checked to only
+               update the preconditioner when `newW == true`.
+           - `Plprev`: the previous `Pl`.
+           - `Prprev`: the previous `Pr`.
+           - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+               Solver-dependent and subject to change.
+         The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+         To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+         which is not used. Additionally, `precs` must supply the dispatch:
+         ```julia
+         Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+         ```
+         which is used in the solver setup phase to construct the integrator
+         type with the preconditioners `(Pl,Pr)`.
+         The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+         is defined as:
+         ```julia
+         DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+         ```)

Rosenbrock Method. 3rd order L-stable Rosenbrock method with 3 internal stages with an embedded strongly A-stable 2nd order method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)
source
OrdinaryDiffEqRosenbrock.ROS3PR — Type
ROS3PR(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `ROS3PR(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```)

Rosenbrock Method. 3nd order stiffly accurate Rosenbrock method with 3 internal stages with B_PR consistent of order 3, which is strongly A-stable with Rinf~=-0.73.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.Scholz4_7 — Type
Scholz4_7(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+                and thus uses the internal ForwardDiff.jl algorithm for the choice.
+            - `standardtag`: Specifies whether to use package-specific tags instead of the
+                ForwardDiff default function-specific tags. For more information, see
+                [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+                Defaults to `Val{true}()`.
+            - `autodiff`: Specifies whether to use automatic differentiation via
+                [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+                differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+                Defaults to `Val{true}()` for automatic differentiation.
+            - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+                `nothing`, which means it will be chosen true/false depending on circumstances
+                of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+            - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+                Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+                `Val{:complex}`.
+            - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+              For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+              `Scholz4_7(linsolve = KLUFactorization()`).
+               When `nothing` is passed, uses `DefaultLinearSolver`.
+            - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+              can be used as a left or right preconditioner.
+              Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+              function where the arguments are defined as:
+                - `W`: the current Jacobian of the nonlinear system. Specified as either
+                    ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                    commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                    representation of the operator. Users can construct the W-matrix on demand
+                    by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                    the `jac_prototype`.
+                - `du`: the current ODE derivative
+                - `u`: the current ODE state
+                - `p`: the ODE parameters
+                - `t`: the current ODE time
+                - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                    the last call to `precs`. It is recommended that this is checked to only
+                    update the preconditioner when `newW == true`.
+                - `Plprev`: the previous `Pl`.
+                - `Prprev`: the previous `Pr`.
+                - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                    Solver-dependent and subject to change.
+              The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+              To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+              which is not used. Additionally, `precs` must supply the dispatch:
+              ```julia
+              Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+              ```
+              which is used in the solver setup phase to construct the integrator
+              type with the preconditioners `(Pl,Pr)`.
+              The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+              is defined as:
+              ```julia
+              DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+              ```)

Rosenbrock Method. 3nd order stiffly accurate Rosenbrock method with 3 internal stages with B_PR consistent of order 3, which is strongly A-stable with Rinf~=-0.73. Convergence with order 4 for the stiff case, but has a poor accuracy.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.ROS34PW1a — Type
ROS34PW1a(; chunk_size = Val{0}(),
+            standardtag = Val{true}(),
+            autodiff = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:central},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW1a(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}

source
OrdinaryDiffEqRosenbrock.ROS34PW1b — Type
ROS34PW1b(; chunk_size = Val{0}(),
+            standardtag = Val{true}(),
+            autodiff = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:central},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW1b(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}

source
OrdinaryDiffEqRosenbrock.ROS34PW2 — Type
ROS34PW2(; chunk_size = Val{0}(),
+           standardtag = Val{true}(),
+           autodiff = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:central},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. A 4th order stiffy accurate Rosenbrock-W method for PDAEs.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}

source
OrdinaryDiffEqRosenbrock.ROS34PW3 — Type
ROS34PW3(; chunk_size = Val{0}(),
+           standardtag = Val{true}(),
+           autodiff = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:central},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. A 4th order strongly A-stable (Rinf~0.63) Rosenbrock-W method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}

source
OrdinaryDiffEqRosenbrock.ROS34PRw — Type
ROS34PRw(; chunk_size = Val{0}(),
+           standardtag = Val{true}(),
+           autodiff = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:central},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. 3rd order stiffly accurate Rosenbrock-Wanner W-method with 4 internal stages, B_PR consistent of order 2. The order of convergence decreases if medium stiff problems are considered.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PRw(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

  • Joachim Rang, Improved traditional Rosenbrock–Wanner methods for stiff ODEs and DAEs, Journal of Computational and Applied Mathematics, https://doi.org/10.1016/j.cam.2015.03.010
source
OrdinaryDiffEqRosenbrock.ROS3PRL — Type
ROS3PRL(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+              and thus uses the internal ForwardDiff.jl algorithm for the choice.
+          - `standardtag`: Specifies whether to use package-specific tags instead of the
+              ForwardDiff default function-specific tags. For more information, see
+              [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+              Defaults to `Val{true}()`.
+          - `autodiff`: Specifies whether to use automatic differentiation via
+              [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+              differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+              Defaults to `Val{true}()` for automatic differentiation.
+          - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+              `nothing`, which means it will be chosen true/false depending on circumstances
+              of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+          - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+              Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+              `Val{:complex}`.
+          - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+            For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+            `ROS3PRL(linsolve = KLUFactorization()`).
+             When `nothing` is passed, uses `DefaultLinearSolver`.
+          - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+            can be used as a left or right preconditioner.
+            Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+            function where the arguments are defined as:
+              - `W`: the current Jacobian of the nonlinear system. Specified as either
+                  ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                  commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                  representation of the operator. Users can construct the W-matrix on demand
+                  by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                  the `jac_prototype`.
+              - `du`: the current ODE derivative
+              - `u`: the current ODE state
+              - `p`: the ODE parameters
+              - `t`: the current ODE time
+              - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                  the last call to `precs`. It is recommended that this is checked to only
+                  update the preconditioner when `newW == true`.
+              - `Plprev`: the previous `Pl`.
+              - `Prprev`: the previous `Pr`.
+              - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                  Solver-dependent and subject to change.
+            The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+            To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+            which is not used. Additionally, `precs` must supply the dispatch:
+            ```julia
+            Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+            ```
+            which is used in the solver setup phase to construct the integrator
+            type with the preconditioners `(Pl,Pr)`.
+            The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+            is defined as:
+            ```julia
+            DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+            ```)

Rosenbrock Method. 3rd order stiffly accurate Rosenbrock method with 4 internal stages, B_PR consistent of order 2 with Rinf=0. The order of convergence decreases if medium stiff problems are considered, but it has good results for very stiff cases.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.ROS3PRL2 — Type
ROS3PRL2(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+               and thus uses the internal ForwardDiff.jl algorithm for the choice.
+           - `standardtag`: Specifies whether to use package-specific tags instead of the
+               ForwardDiff default function-specific tags. For more information, see
+               [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+               Defaults to `Val{true}()`.
+           - `autodiff`: Specifies whether to use automatic differentiation via
+               [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+               differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+               Defaults to `Val{true}()` for automatic differentiation.
+           - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+               `nothing`, which means it will be chosen true/false depending on circumstances
+               of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+           - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+               Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+               `Val{:complex}`.
+           - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+             For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+             `ROS3PRL2(linsolve = KLUFactorization()`).
+              When `nothing` is passed, uses `DefaultLinearSolver`.
+           - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+             can be used as a left or right preconditioner.
+             Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+             function where the arguments are defined as:
+               - `W`: the current Jacobian of the nonlinear system. Specified as either
+                   ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                   commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                   representation of the operator. Users can construct the W-matrix on demand
+                   by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                   the `jac_prototype`.
+               - `du`: the current ODE derivative
+               - `u`: the current ODE state
+               - `p`: the ODE parameters
+               - `t`: the current ODE time
+               - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                   the last call to `precs`. It is recommended that this is checked to only
+                   update the preconditioner when `newW == true`.
+               - `Plprev`: the previous `Pl`.
+               - `Prprev`: the previous `Pr`.
+               - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                   Solver-dependent and subject to change.
+             The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+             To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+             which is not used. Additionally, `precs` must supply the dispatch:
+             ```julia
+             Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+             ```
+             which is used in the solver setup phase to construct the integrator
+             type with the preconditioners `(Pl,Pr)`.
+             The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+             is defined as:
+             ```julia
+             DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+             ```)

Rosenbrock Method. 3rd order stiffly accurate Rosenbrock method with 4 internal stages, B_PR consistent of order 3. The order of convergence does NOT decreases if medium stiff problems are considered as it does for ROS3PRL.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • Rang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0
source
OrdinaryDiffEqRosenbrock.ROK4a — Type
ROK4a(; chunk_size = Val{0}(),
+        standardtag = Val{true}(),
+        autodiff = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:central},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS)

Rosenbrock-Wanner Method. 4rd order L-stable Rosenbrock-Krylov method with 4 internal stages, with a 3rd order embedded method which is strongly A-stable with Rinf~=0.55. (when using exact Jacobians)

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROK4a(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing

References

  • Tranquilli, Paul and Sandu, Adrian (2014): Rosenbrock–Krylov Methods for Large Systems of Differential Equations https://doi.org/10.1137/130923336
source
OrdinaryDiffEqRosenbrock.RosShamp4 — Type
RosShamp4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+                and thus uses the internal ForwardDiff.jl algorithm for the choice.
+            - `standardtag`: Specifies whether to use package-specific tags instead of the
+                ForwardDiff default function-specific tags. For more information, see
+                [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+                Defaults to `Val{true}()`.
+            - `autodiff`: Specifies whether to use automatic differentiation via
+                [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+                differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+                Defaults to `Val{true}()` for automatic differentiation.
+            - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+                `nothing`, which means it will be chosen true/false depending on circumstances
+                of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+            - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+                Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+                `Val{:complex}`.
+            - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+              For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+              `RosShamp4(linsolve = KLUFactorization()`).
+               When `nothing` is passed, uses `DefaultLinearSolver`.
+            - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+              can be used as a left or right preconditioner.
+              Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+              function where the arguments are defined as:
+                - `W`: the current Jacobian of the nonlinear system. Specified as either
+                    ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                    commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                    representation of the operator. Users can construct the W-matrix on demand
+                    by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                    the `jac_prototype`.
+                - `du`: the current ODE derivative
+                - `u`: the current ODE state
+                - `p`: the ODE parameters
+                - `t`: the current ODE time
+                - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                    the last call to `precs`. It is recommended that this is checked to only
+                    update the preconditioner when `newW == true`.
+                - `Plprev`: the previous `Pl`.
+                - `Prprev`: the previous `Pr`.
+                - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                    Solver-dependent and subject to change.
+              The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+              To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+              which is not used. Additionally, `precs` must supply the dispatch:
+              ```julia
+              Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+              ```
+              which is used in the solver setup phase to construct the integrator
+              type with the preconditioners `(Pl,Pr)`.
+              The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+              is defined as:
+              ```julia
+              DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+              ```)

Rosenbrock Method. An A-stable 4th order Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver

References

  • L. F. Shampine, Implementation of Rosenbrock Methods, ACM Transactions on Mathematical Software (TOMS), 8: 2, 93-113. doi:10.1145/355993.355994
source
OrdinaryDiffEqRosenbrock.Veldd4 — Type
Veldd4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `Veldd4(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 4th order D-stable Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • van Veldhuizen, D-stability and Kaps-Rentrop-methods, M. Computing (1984) 32: 229. doi:10.1007/BF02243574
source
OrdinaryDiffEqRosenbrock.Velds4 — Type
Velds4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+             and thus uses the internal ForwardDiff.jl algorithm for the choice.
+         - `standardtag`: Specifies whether to use package-specific tags instead of the
+             ForwardDiff default function-specific tags. For more information, see
+             [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+             Defaults to `Val{true}()`.
+         - `autodiff`: Specifies whether to use automatic differentiation via
+             [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+             differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+             Defaults to `Val{true}()` for automatic differentiation.
+         - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+             `nothing`, which means it will be chosen true/false depending on circumstances
+             of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+         - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+             Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+             `Val{:complex}`.
+         - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+           For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+           `Velds4(linsolve = KLUFactorization()`).
+            When `nothing` is passed, uses `DefaultLinearSolver`.
+         - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+           can be used as a left or right preconditioner.
+           Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+           function where the arguments are defined as:
+             - `W`: the current Jacobian of the nonlinear system. Specified as either
+                 ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                 commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                 representation of the operator. Users can construct the W-matrix on demand
+                 by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                 the `jac_prototype`.
+             - `du`: the current ODE derivative
+             - `u`: the current ODE state
+             - `p`: the ODE parameters
+             - `t`: the current ODE time
+             - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                 the last call to `precs`. It is recommended that this is checked to only
+                 update the preconditioner when `newW == true`.
+             - `Plprev`: the previous `Pl`.
+             - `Prprev`: the previous `Pr`.
+             - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                 Solver-dependent and subject to change.
+           The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+           To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+           which is not used. Additionally, `precs` must supply the dispatch:
+           ```julia
+           Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+           ```
+           which is used in the solver setup phase to construct the integrator
+           type with the preconditioners `(Pl,Pr)`.
+           The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+           is defined as:
+           ```julia
+           DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+           ```
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 4th order A-stable Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • van Veldhuizen, D-stability and Kaps-Rentrop-methods, M. Computing (1984) 32: 229. doi:10.1007/BF02243574
source
OrdinaryDiffEqRosenbrock.GRK4T — Type
GRK4T(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+            and thus uses the internal ForwardDiff.jl algorithm for the choice.
+        - `standardtag`: Specifies whether to use package-specific tags instead of the
+            ForwardDiff default function-specific tags. For more information, see
+            [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+            Defaults to `Val{true}()`.
+        - `autodiff`: Specifies whether to use automatic differentiation via
+            [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+            differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+            Defaults to `Val{true}()` for automatic differentiation.
+        - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+            `nothing`, which means it will be chosen true/false depending on circumstances
+            of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+        - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+            Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+            `Val{:complex}`.
+        - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+          For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+          `GRK4T(linsolve = KLUFactorization()`).
+           When `nothing` is passed, uses `DefaultLinearSolver`.
+        - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+          can be used as a left or right preconditioner.
+          Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+          function where the arguments are defined as:
+            - `W`: the current Jacobian of the nonlinear system. Specified as either
+                ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                representation of the operator. Users can construct the W-matrix on demand
+                by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                the `jac_prototype`.
+            - `du`: the current ODE derivative
+            - `u`: the current ODE state
+            - `p`: the ODE parameters
+            - `t`: the current ODE time
+            - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                the last call to `precs`. It is recommended that this is checked to only
+                update the preconditioner when `newW == true`.
+            - `Plprev`: the previous `Pl`.
+            - `Prprev`: the previous `Pr`.
+            - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                Solver-dependent and subject to change.
+          The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+          To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+          which is not used. Additionally, `precs` must supply the dispatch:
+          ```julia
+          Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+          ```
+          which is used in the solver setup phase to construct the integrator
+          type with the preconditioners `(Pl,Pr)`.
+          The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+          is defined as:
+          ```julia
+          DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+          ```
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. An efficient 4th order Rosenbrock method.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Kaps, P. & Rentrop, Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. P. Numer. Math. (1979) 33: 55. doi:10.1007/BF01396495
source
OrdinaryDiffEqRosenbrock.GRK4A — Type
GRK4A(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+            and thus uses the internal ForwardDiff.jl algorithm for the choice.
+        - `standardtag`: Specifies whether to use package-specific tags instead of the
+            ForwardDiff default function-specific tags. For more information, see
+            [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+            Defaults to `Val{true}()`.
+        - `autodiff`: Specifies whether to use automatic differentiation via
+            [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+            differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+            Defaults to `Val{true}()` for automatic differentiation.
+        - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+            `nothing`, which means it will be chosen true/false depending on circumstances
+            of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+        - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+            Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+            `Val{:complex}`.
+        - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+          For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+          `GRK4A(linsolve = KLUFactorization()`).
+           When `nothing` is passed, uses `DefaultLinearSolver`.
+        - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+          can be used as a left or right preconditioner.
+          Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+          function where the arguments are defined as:
+            - `W`: the current Jacobian of the nonlinear system. Specified as either
+                ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                representation of the operator. Users can construct the W-matrix on demand
+                by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                the `jac_prototype`.
+            - `du`: the current ODE derivative
+            - `u`: the current ODE state
+            - `p`: the ODE parameters
+            - `t`: the current ODE time
+            - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                the last call to `precs`. It is recommended that this is checked to only
+                update the preconditioner when `newW == true`.
+            - `Plprev`: the previous `Pl`.
+            - `Prprev`: the previous `Pr`.
+            - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                Solver-dependent and subject to change.
+          The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+          To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+          which is not used. Additionally, `precs` must supply the dispatch:
+          ```julia
+          Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+          ```
+          which is used in the solver setup phase to construct the integrator
+          type with the preconditioners `(Pl,Pr)`.
+          The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+          is defined as:
+          ```julia
+          DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+          ```
+        step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. An A-stable 4th order Rosenbrock method. Essentially "anti-L-stable" but efficient.

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • Kaps, P. & Rentrop, Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. P. Numer. Math. (1979) 33: 55. doi:10.1007/BF01396495
source
OrdinaryDiffEqRosenbrock.Ros4LStab — Type
Ros4LStab(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`
+                and thus uses the internal ForwardDiff.jl algorithm for the choice.
+            - `standardtag`: Specifies whether to use package-specific tags instead of the
+                ForwardDiff default function-specific tags. For more information, see
+                [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).
+                Defaults to `Val{true}()`.
+            - `autodiff`: Specifies whether to use automatic differentiation via
+                [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite
+                differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).
+                Defaults to `Val{true}()` for automatic differentiation.
+            - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to
+                `nothing`, which means it will be chosen true/false depending on circumstances
+                of the solver, such as whether a Krylov subspace method is used for `linsolve`.
+            - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.
+                Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and
+                `Val{:complex}`.
+            - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.
+              For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify
+              `Ros4LStab(linsolve = KLUFactorization()`).
+               When `nothing` is passed, uses `DefaultLinearSolver`.
+            - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)
+              can be used as a left or right preconditioner.
+              Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`
+              function where the arguments are defined as:
+                - `W`: the current Jacobian of the nonlinear system. Specified as either
+                    ``I - \gamma J`` or ``I/\gamma - J`` depending on the algorithm. This will
+                    commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy
+                    representation of the operator. Users can construct the W-matrix on demand
+                    by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching
+                    the `jac_prototype`.
+                - `du`: the current ODE derivative
+                - `u`: the current ODE state
+                - `p`: the ODE parameters
+                - `t`: the current ODE time
+                - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since
+                    the last call to `precs`. It is recommended that this is checked to only
+                    update the preconditioner when `newW == true`.
+                - `Plprev`: the previous `Pl`.
+                - `Prprev`: the previous `Pr`.
+                - `solverdata`: Optional extra data the solvers can give to the `precs` function.
+                    Solver-dependent and subject to change.
+              The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.
+              To specify one-sided preconditioning, simply return `nothing` for the preconditioner
+              which is not used. Additionally, `precs` must supply the dispatch:
+              ```julia
+              Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
+              ```
+              which is used in the solver setup phase to construct the integrator
+              type with the preconditioners `(Pl,Pr)`.
+              The default is `precs=DEFAULT_PRECS` where the default preconditioner function
+              is defined as:
+              ```julia
+              DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
+              ```
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Rosenbrock Method. A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant

Keyword Arguments

  • chunk_size: TBD
  • standardtag: TBD
  • autodiff: boolean to control if the Jacobian should be computed via AD or not
  • concrete_jac: function of the form jac!(J, u, p, t)
  • diff_type: TBD
  • linsolve: custom solver for the inner linear systems
  • precs: custom preconditioner for the inner linear solver
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

  • E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)
source
diff --git a/dev/implicit/SDIRK/index.html b/dev/implicit/SDIRK/index.html new file mode 100644 index 0000000000..21bf7b5f76 --- /dev/null +++ b/dev/implicit/SDIRK/index.html @@ -0,0 +1,299 @@ + +OrdinaryDiffEqSDIRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqSDIRK

This article is a stub.

Installation

To be able to access the solvers in OrdinaryDiffEqSDIRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqSDIRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqSDIRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, PDIRK44())

Full list of solvers

OrdinaryDiffEqSDIRK.ImplicitEuler — Type
ImplicitEuler(; chunk_size = Val{0}(),
+                autodiff = true,
+                standardtag = Val{true}(),
+                concrete_jac = nothing,
+                diff_type = Val{:forward},
+                linsolve = nothing,
+                precs = DEFAULT_PRECS,
+                nlsolve = NLNewton(),
+                extrapolant = :constant,
+                controller = :PI,
+                step_limiter! = trivial_limiter!)

SDIRK Method. A 1st order implicit solver. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEuler(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@book{wanner1996solving, title={Solving ordinary differential equations II}, author={Wanner, Gerhard and Hairer, Ernst}, volume={375}, year={1996}, publisher={Springer Berlin Heidelberg New York}}

source
OrdinaryDiffEqSDIRK.ImplicitMidpoint — Type
ImplicitMidpoint(; chunk_size = Val{0}(),
+                   autodiff = true,
+                   standardtag = Val{true}(),
+                   concrete_jac = nothing,
+                   diff_type = Val{:forward},
+                   linsolve = nothing,
+                   precs = DEFAULT_PRECS,
+                   nlsolve = NLNewton(),
+                   extrapolant = :linear,
+                   step_limiter! = trivial_limiter!)

SDIRK Method. A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitMidpoint(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@book{wanner1996solving, title={Solving ordinary differential equations II}, author={Wanner, Gerhard and Hairer, Ernst}, volume={375}, year={1996}, publisher={Springer Berlin Heidelberg New York}}

source
OrdinaryDiffEqSDIRK.Trapezoid — Type
Trapezoid(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            nlsolve = NLNewton(),
+            extrapolant = :linear,
+            controller = :PI,
+            step_limiter! = trivial_limiter!)

SDIRK Method. Second order A-stable symmetric ESDIRK method. "Almost symplectic" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences approximation to the second derivative terms in the local truncation error estimate (the SPICE approximation strategy).

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Trapezoid(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Andre Vladimirescu. 1994. The Spice Book. John Wiley & Sons, Inc., New York, NY, USA.

source
OrdinaryDiffEqSDIRK.TRBDF2 — Type
TRBDF2(; chunk_size = Val{0}(),
+         autodiff = true,
+         standardtag = Val{true}(),
+         concrete_jac = nothing,
+         diff_type = Val{:forward},
+         linsolve = nothing,
+         precs = DEFAULT_PRECS,
+         nlsolve = NLNewton(),
+         smooth_est = true,
+         extrapolant = :linear,
+         controller = :PI,
+         step_limiter! = trivial_limiter!)

SDIRK Method. A second order A-B-L-S-stable one-step ESDIRK method. Includes stiffness-robust error estimates for accurate adaptive timestepping, smoothed derivatives for highly stiff and oscillatory problems.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify TRBDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hosea1996analysis, title={Analysis and implementation of TR-BDF2}, author={Hosea, ME and Shampine, LF}, journal={Applied Numerical Mathematics}, volume={20}, number={1-2}, pages={21–37}, year={1996}, publisher={Elsevier}

source
OrdinaryDiffEqSDIRK.SDIRK2 — Type
SDIRK2(; chunk_size = Val{0}(),
+         autodiff = true,
+         standardtag = Val{true}(),
+         concrete_jac = nothing,
+         diff_type = Val{:forward},
+         linsolve = nothing,
+         precs = DEFAULT_PRECS,
+         nlsolve = NLNewton(),
+         smooth_est = true,
+         extrapolant = :linear,
+         controller = :PI,
+         step_limiter! = trivial_limiter!)

SDIRK Method. SDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SDIRK2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM}}

source
OrdinaryDiffEqSDIRK.SDIRK22 — Type
SDIRK22(; chunk_size = Val{0}(),
+          autodiff = true,
+          standardtag = Val{true}(),
+          concrete_jac = nothing,
+          diff_type = Val{:forward},
+          linsolve = nothing,
+          precs = DEFAULT_PRECS,
+          nlsolve = NLNewton(),
+          smooth_est = true,
+          extrapolant = :linear,
+          controller = :PI,
+          step_limiter! = trivial_limiter!)

SDIRK Method. Description TBD

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SDIRK22(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@techreport{kennedy2016diagonally, title={Diagonally implicit Runge-Kutta methods for ordinary differential equations. A review}, author={Kennedy, Christopher A and Carpenter, Mark H}, year={2016}}

source
OrdinaryDiffEqSDIRK.SSPSDIRK2 — Type
SSPSDIRK2(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            nlsolve = NLNewton(),
+            smooth_est = true,
+            extrapolant = :constant,
+            controller = :PI)

SDIRK Method. SSPSDIRK is an SSP-optimized SDIRK method, so it's an implicit SDIRK method for handling stiffness but if the dt is below the SSP coefficient * dt, then the SSP property of the SSP integrators (the other page) is satisified. As such this is a method which is expected to be good on advection-dominated cases where an explicit SSP integrator would be used, but where reaction equations are sufficient stiff to justify implicit integration.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SSPSDIRK2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD

References

@article{ketcheson2009optimal, title={Optimal implicit strong stability preserving Runge–Kutta methods}, author={Ketcheson, David I and Macdonald, Colin B and Gottlieb, Sigal}, journal={Applied Numerical Mathematics}, volume={59}, number={2}, pages={373–392}, year={2009}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.Kvaerno3 — Type
Kvaerno3(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 3rd order ESDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}

source
OrdinaryDiffEqSDIRK.KenCarp3 — Type
KenCarp3(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}

source
OrdinaryDiffEqSDIRK.CFNLIRK3 — Type
CFNLIRK3(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Third order method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify CFNLIRK3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{calvo2001linearly, title={Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations}, author={Calvo, MP and De Frutos, J and Novo, J}, journal={Applied Numerical Mathematics}, volume={37}, number={4}, pages={535–549}, year={2001}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.Cash4 — Type
Cash4(; chunk_size = Val{0}(),
+        autodiff = true,
+        standardtag = Val{true}(),
+        concrete_jac = nothing,
+        diff_type = Val{:forward},
+        linsolve = nothing,
+        precs = DEFAULT_PRECS,
+        nlsolve = NLNewton(),
+        smooth_est = true,
+        extrapolant = :linear,
+        controller = :PI,
+        embedding = 3)

SDIRK Method. An A-L stable 4th order SDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Cash4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • embedding: TBD

References

@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM}}

source
OrdinaryDiffEqSDIRK.SFSDIRK4 — Type
SFSDIRK4(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Method of order 4.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.SFSDIRK5 — Type
SFSDIRK5(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Method of order 5.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.SFSDIRK6 — Type
SFSDIRK6(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Method of order 6.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK6(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.SFSDIRK7 — Type
SFSDIRK7(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Method of order 7.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK7(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.SFSDIRK8 — Type
SFSDIRK8(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           extrapolant = :linear)

SDIRK Method. Method of order 8.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK8(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD

References

@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.Hairer4 — Type
Hairer4(; chunk_size = Val{0}(),
+          autodiff = true,
+          standardtag = Val{true}(),
+          concrete_jac = nothing,
+          diff_type = Val{:forward},
+          linsolve = nothing,
+          precs = DEFAULT_PRECS,
+          nlsolve = NLNewton(),
+          smooth_est = true,
+          extrapolant = :linear,
+          controller = :PI)

SDIRK Method. An A-L stable 4th order SDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Hairer4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD

References

E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

source
OrdinaryDiffEqSDIRK.Hairer42 — Type
Hairer42(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI)

SDIRK Method. An A-L stable 4th order SDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Hairer42(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD

References

E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

source
OrdinaryDiffEqSDIRK.Kvaerno4 — Type
Kvaerno4(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 4th order ESDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter: TBD

References

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}

source
OrdinaryDiffEqSDIRK.Kvaerno5 — Type
Kvaerno5(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 5th order ESDIRK method.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter: TBD

References

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}

source

IMEX SDIRK

OrdinaryDiffEqSDIRK.KenCarp4 — Type
KenCarp4(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 4th order ESDIRK method with splitting.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter: TBD

References

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}

source
OrdinaryDiffEqSDIRK.KenCarp47 — Type
KenCarp47(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            nlsolve = NLNewton(),
+            smooth_est = true,
+            extrapolant = :linear,
+            controller = :PI)

SDIRK Method. An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp47(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD

References

@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.KenCarp5 — Type
KenCarp5(; chunk_size = Val{0}(),
+           autodiff = true,
+           standardtag = Val{true}(),
+           concrete_jac = nothing,
+           diff_type = Val{:forward},
+           linsolve = nothing,
+           precs = DEFAULT_PRECS,
+           nlsolve = NLNewton(),
+           smooth_est = true,
+           extrapolant = :linear,
+           controller = :PI,
+           step_limiter! = trivial_limiter!)

SDIRK Method. An A-L stable stiffly-accurate 5th order ESDIRK method with splitting.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD
  • step_limiter: TBD

References

@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}

source
OrdinaryDiffEqSDIRK.KenCarp58 — Type
KenCarp58(; chunk_size = Val{0}(),
+            autodiff = true,
+            standardtag = Val{true}(),
+            concrete_jac = nothing,
+            diff_type = Val{:forward},
+            linsolve = nothing,
+            precs = DEFAULT_PRECS,
+            nlsolve = NLNewton(),
+            smooth_est = true,
+            extrapolant = :linear,
+            controller = :PI)

SDIRK Method. An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp58(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • smooth_est: TBD
  • extrapolant: TBD
  • controller: TBD

References

@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier}}

source
OrdinaryDiffEqSDIRK.ESDIRK54I8L2SA — Type
ESDIRK54I8L2SA(; chunk_size = Val{0}(),
+                 autodiff = true,
+                 standardtag = Val{true}(),
+                 concrete_jac = nothing,
+                 diff_type = Val{:forward},
+                 linsolve = nothing,
+                 precs = DEFAULT_PRECS,
+                 nlsolve = NLNewton(),
+                 extrapolant = :linear,
+                 controller = :PI)

SDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK54I8L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD

References

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK436L2SA2 — Type
ESDIRK436L2SA2(; chunk_size = Val{0}(),
+                 autodiff = true,
+                 standardtag = Val{true}(),
+                 concrete_jac = nothing,
+                 diff_type = Val{:forward},
+                 linsolve = nothing,
+                 precs = DEFAULT_PRECS,
+                 nlsolve = NLNewton(),
+                 extrapolant = :linear,
+                 controller = :PI)

SDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK436L2SA2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD

References

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK437L2SA — Type
ESDIRK437L2SA(; chunk_size = Val{0}(),
+                autodiff = true,
+                standardtag = Val{true}(),
+                concrete_jac = nothing,
+                diff_type = Val{:forward},
+                linsolve = nothing,
+                precs = DEFAULT_PRECS,
+                nlsolve = NLNewton(),
+                extrapolant = :linear,
+                controller = :PI)

SDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK437L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD

References

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK547L2SA2 — Type
ESDIRK547L2SA2(; chunk_size = Val{0}(),
+                 autodiff = true,
+                 standardtag = Val{true}(),
+                 concrete_jac = nothing,
+                 diff_type = Val{:forward},
+                 linsolve = nothing,
+                 precs = DEFAULT_PRECS,
+                 nlsolve = NLNewton(),
+                 extrapolant = :linear,
+                 controller = :PI)

SDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK547L2SA2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD

References

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
OrdinaryDiffEqSDIRK.ESDIRK659L2SA — Type
ESDIRK659L2SA(; chunk_size = Val{0}(),
+                autodiff = true,
+                standardtag = Val{true}(),
+                concrete_jac = nothing,
+                diff_type = Val{:forward},
+                linsolve = nothing,
+                precs = DEFAULT_PRECS,
+                nlsolve = NLNewton(),
+                extrapolant = :linear,
+                controller = :PI)

SDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context. Currently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details.

Keyword Arguments

  • chunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.
  • autodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • diff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK659L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • nlsolve: TBD
    • extrapolant: TBD
  • controller: TBD

References

@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }

source
diff --git a/dev/implicit/StabalizedIRK/index.html b/dev/implicit/StabalizedIRK/index.html new file mode 100644 index 0000000000..8797b76267 --- /dev/null +++ b/dev/implicit/StabalizedIRK/index.html @@ -0,0 +1,15 @@ + +OrdinaryDiffEqStabalizedIRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqStabalizedIRK

Stabilized Explicit Runge-Kutta methods, like Runge-Kutta-Chebyshev methods and ROCK methods are explicit methods which chain together many stages in a specific way to get large stability regions. they are made in such a way to converge to a large stability region, and thus suitable to stiff equations. However, they converge to having a large stability region in the direction of the negative real axis, and thus are only stable on a subset of stiff equations which are not dominated by large complex eigenvalues in the Jacobian.

Stabilized implicit methods try to mitigate this problem by being an IMEX type scheme, requiring a SplitODEProblem where the splitting is designed to treat the large complex eigenvalues implicitly while treating the large real eigenvalues using a fast explicit stabilized RK type of method.

These methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example

`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`

Installation

To be able to access the solvers in OrdinaryDiffEqStabalizedIRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqStabalizedIRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqStabilizedIRK
+A = randn(20, 20)
+B = randn(20, 20)
+f1 = (u, p, t) -> A * u
+f2 = (u, p, t) -> B * u
+u0 = randn(20, 1)
+tspan = (0.0, 1.0)
+prob = SplitODEProblem(f1, f2, u0, tspan)
+sol = solve(prob, IRKC())

Full list of solvers

OrdinaryDiffEqStabilizedIRK.IRKC — Type
IRKC(; eigen_est = nothing)

Stabalized Implicit Runge Kutta method. Implicit Runge-Kutta-Chebyshev method.

Keyword Arguments

  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

REF TBD

source
diff --git a/dev/implicit/StabalizedRK/index.html b/dev/implicit/StabalizedRK/index.html new file mode 100644 index 0000000000..d34b7090d8 --- /dev/null +++ b/dev/implicit/StabalizedRK/index.html @@ -0,0 +1,22 @@ + +OrdinaryDiffEqStabalizedRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqStabalizedRK

Explicit stabilized methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example

`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`

Installation

To be able to access the solvers in OrdinaryDiffEqStabalizedRK, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqStabalizedRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqStabalizedRK
+
+function lorenz!(du, u, p, t)
+    du[1] = 10.0 * (u[2] - u[1])
+    u[2] = u[1] * (28.0 - u[3]) - u[2]
+    du[3] = u[1] * u[2] - (8 / 3) * u[3]
+end
+u0 = [1.0; 0.0; 0.0]
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz!, u0, tspan)
+sol = solve(prob, ROCK4())

Full list of solvers

OrdinaryDiffEqStabilizedRK.ROCK2 — Type
ROCK2(; min_stages = 0,
+        max_stages = 200,
+        eigen_est = nothing)

Stabilized Explicit Method. Second order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

Keyword Arguments

  • min_stages: The minimum degree of the Chebyshev polynomial.
  • max_stages: The maximumdegree of the Chebyshev polynomial.
  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

Assyr Abdulle, Alexei A. Medovikov. Second Order Chebyshev Methods based on Orthogonal Polynomials. Numerische Mathematik, 90 (1), pp 1-18, 2001. doi: https://dx.doi.org/10.1007/s002110100292

source
OrdinaryDiffEqStabilizedRK.ROCK4 — Type
ROCK4(; min_stages = 0,
+        max_stages = 152,
+        eigen_est = nothing)

Stabilized Explicit Method. Fourth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

Keyword Arguments

  • min_stages: The minimum degree of the Chebyshev polynomial.
  • max_stages: The maximumdegree of the Chebyshev polynomial.
  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

Assyr Abdulle. Fourth Order Chebyshev Methods With Recurrence Relation. 2002 Society for Industrial and Applied Mathematics Journal on Scientific Computing, 23(6), pp 2041-2054, 2001. doi: https://doi.org/10.1137/S1064827500379549

source
OrdinaryDiffEqStabilizedRK.RKC — Type
RKC(; eigen_est = nothing)

Stabilized Explicit Method. Second order method. Exhibits high stability for real eigenvalues.

Keyword Arguments

  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

B. P. Sommeijer, L. F. Shampine, J. G. Verwer. RKC: An Explicit Solver for Parabolic PDEs, Journal of Computational and Applied Mathematics, 88(2), pp 315-326, 1998. doi: https://doi.org/10.1016/S0377-0427(97)00219-7

source
OrdinaryDiffEqStabilizedRK.SERK2 — Type
SERK2(; controller = :PI
+        eigen_est = nothing)

Stabilized Explicit Method. Second order method.

Keyword Arguments

  • controller: TBD
  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

@article{kleefeld2013serk2v2, title={SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems}, author={Kleefeld, B and Martin-Vaquero, J}, journal={Numerical Methods for Partial Differential Equations}, volume={29}, number={1}, pages={170–185}, year={2013}, publisher={Wiley Online Library}}

source
OrdinaryDiffEqStabilizedRK.ESERK4 — Type
ESERK4(; eigen_est = nothing)

Stabilized Explicit Method. Fourth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

Keyword Arguments

  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

J. Martín-Vaquero, B. Kleefeld. Extrapolated stabilized explicit Runge-Kutta methods, Journal of Computational Physics, 326, pp 141-155, 2016. doi: https://doi.org/10.1016/j.jcp.2016.08.042.

source
OrdinaryDiffEqStabilizedRK.ESERK5 — Type
ESERK5(; eigen_est = nothing)

Stabilized Explicit Method. Fifth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

Keyword Arguments

  • eigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

References

J. Martín-Vaquero, A. Kleefeld. ESERK5: A fifth-order extrapolated stabilized explicit Runge-Kutta method, Journal of Computational and Applied Mathematics, 356, pp 22-36, 2019. doi: https://doi.org/10.1016/j.cam.2019.01.040.

source
diff --git a/dev/index.html b/dev/index.html index f18dee948f..1a3b9d4a29 100644 --- a/dev/index.html +++ b/dev/index.html @@ -3,7 +3,7 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

OrdinaryDiffEq.jl

OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.

Installation

Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:

import Pkg;
+
GitHub
OrdinaryDiffEq.jl
OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.
Installation
Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:
import Pkg;
 Pkg.add("OrdinaryDiffEq");
Reproducibility
The documentation of this SciML package was built using these direct dependencies,
Status `~/work/OrdinaryDiffEq.jl/OrdinaryDiffEq.jl/docs/Project.toml`
   [e30172f5] Documenter v1.6.0
   [1dea7af3] OrdinaryDiffEq v6.88.1 `~/work/OrdinaryDiffEq.jl/OrdinaryDiffEq.jl`
and using this machine and Julia version.
Julia Version 1.10.4
@@ -224,6 +224,4 @@
   [8e850b90] libblastrampoline_jll v5.8.0+1
   [8e850ede] nghttp2_jll v1.52.0+1
   [3f19e933] p7zip_jll v17.4.0+2
-Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
You can also download the 
-"https://github.com/SciML/OrdinaryDiffEq.jl/tree/gh-pages/v6.88.1/assets/Manifest.toml"
">manifest file and the -"https://github.com/SciML/OrdinaryDiffEq.jl/tree/gh-pages/v6.88.1/assets/Project.toml"">project file.
+Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.

diff --git a/dev/misc/index.html b/dev/misc/index.html index 3e0e2e5c75..2442a5d639 100644 --- a/dev/misc/index.html +++ b/dev/misc/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub
Missing docstring.

Missing docstring for LinearExponential. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SplitEuler. Check Documenter's build log for details.

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Missing docstring for CompositeAlgorithm. Check Documenter's build log for details.

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Missing docstring for PDIRK44. Check Documenter's build log for details.

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GitHub
Missing docstring.

Missing docstring for SplitEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CompositeAlgorithm. Check Documenter's build log for details.

diff --git a/dev/nonstiff/explicit_extrapolation/index.html b/dev/nonstiff/explicit_extrapolation/index.html deleted file mode 100644 index d81b02d1b7..0000000000 --- a/dev/nonstiff/explicit_extrapolation/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Explicit Extrapolation Methods · OrdinaryDiffEq.jl
GitHub

Explicit Extrapolation Methods

diff --git a/dev/nonstiff/explicitrk/index.html b/dev/nonstiff/explicitrk/index.html deleted file mode 100644 index 3fba20b67f..0000000000 --- a/dev/nonstiff/explicitrk/index.html +++ /dev/null @@ -1,17 +0,0 @@ - -Explicit Runge-Kutta Methods · OrdinaryDiffEq.jl
GitHub

Explicit Runge-Kutta Methods

With the help of FastBroadcast.jl, we can use threaded parallelism to reduce compute time for all of the explicit Runge-Kutta methods! The thread option determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False(), default) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads. When we call solve(prob, alg(thread=OrdinaryDiffEq.True())), we can turn on the multithreading option to achieve acceleration (for sufficiently large problems).

Standard Explicit Runge-Kutta Methods

Missing docstring.

Missing docstring for Heun. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Ralston. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Midpoint. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RK4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RKM. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MSRK5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MSRK6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Anas5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RKO65. Check Documenter's build log for details.

Missing docstring.

Missing docstring for OwrenZen3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for OwrenZen4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for OwrenZen5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for BS3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for DP5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Tsit5. Check Documenter's build log for details.

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Missing docstring for DP8. Check Documenter's build log for details.

Missing docstring.

Missing docstring for TanYam7. Check Documenter's build log for details.

Missing docstring.

Missing docstring for TsitPap8. Check Documenter's build log for details.

OrdinaryDiffEqFeagin.Feagin10 — Type

@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \& Scientific Computations} }

Feagin10: Explicit Runge-Kutta Method Feagin's 10th-order Runge-Kutta method.

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OrdinaryDiffEqFeagin.Feagin12 — Type

@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \& Scientific Computations} }

Feagin12: Explicit Runge-Kutta Method Feagin's 12th-order Runge-Kutta method.

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OrdinaryDiffEqFeagin.Feagin14 — Type

Feagin, T., “An Explicit Runge-Kutta Method of Order Fourteen,” Numerical Algorithms, 2009

Feagin14: Explicit Runge-Kutta Method Feagin's 14th-order Runge-Kutta method.

source
Missing docstring.

Missing docstring for FRK65. Check Documenter's build log for details.

Missing docstring.

Missing docstring for PFRK87. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Stepanov5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SIR54. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Alshina2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Alshina3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Alshina6. Check Documenter's build log for details.

Lazy Interpolation Explicit Runge-Kutta Methods

Missing docstring.

Missing docstring for BS5. Check Documenter's build log for details.

OrdinaryDiffEqVerner.Vern6 — Type
Vern6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 6/5 Runge-Kutta method. (lazy 6th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

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OrdinaryDiffEqVerner.Vern7 — Type
Vern7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 7/6 Runge-Kutta method. (lazy 7th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

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OrdinaryDiffEqVerner.Vern8 — Type
Vern8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 8/7 Runge-Kutta method. (lazy 8th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

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OrdinaryDiffEqVerner.Vern9 — Type
Vern9(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)

Explicit Runge-Kutta Method. Verner's “Most Efficient” 9/8 Runge-Kutta method. (lazy9th order interpolant).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • lazy: determines if the lazy interpolant is used.

References

@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }

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Fixed Timestep Only Explicit Runge-Kutta Methods

Missing docstring.

Missing docstring for Euler. Check Documenter's build log for details.

OrdinaryDiffEqLowStorageRK.RK46NL — Type
RK46NL(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-         step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Julien Berland, Christophe Bogey, Christophe Bailly. Low-Dissipation and Low-Dispersion Fourth-Order Runge-Kutta Algorithm. Computers & Fluids, 35(10), pp 1459-1463, 2006. doi: https://doi.org/10.1016/j.compfluid.2005.04.003

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OrdinaryDiffEqLowStorageRK.ORK256 — Type
ORK256(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-         williamson_condition = true)

Explicit Runge-Kutta Method. A second-order, five-stage explicit Runge-Kutta method for wave propagation equations. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Matteo Bernardini, Sergio Pirozzoli. A General Strategy for the Optimization of Runge-Kutta Schemes for Wave Propagation Phenomena. Journal of Computational Physics, 228(11), pp 4182-4199, 2009. doi: https://doi.org/10.1016/j.jcp.2009.02.032

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Parallel Explicit Runge-Kutta Methods

Missing docstring.

Missing docstring for KuttaPRK2p5. Check Documenter's build log for details.

diff --git a/dev/nonstiff/lowstorage_ssprk/index.html b/dev/nonstiff/lowstorage_ssprk/index.html deleted file mode 100644 index 0178f170dc..0000000000 --- a/dev/nonstiff/lowstorage_ssprk/index.html +++ /dev/null @@ -1,76 +0,0 @@ - -PDE-Specialized Explicit Runge-Kutta Methods · OrdinaryDiffEq.jl
GitHub

PDE-Specialized Explicit Runge-Kutta Methods

Low Storage Explicit Runge-Kutta Methods

OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54 — Type
CarpenterKennedy2N54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                       step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                       williamson_condition = true)

Explicit Runge-Kutta Method. A fourth-order, five-stage explicit low-storage method of Carpenter and Kennedy (free 3rd order Hermite interpolant). Fixed timestep only. Designed for hyperbolic PDEs (stability properties).

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

@article{carpenter1994fourth, title={Fourth-order 2N-storage Runge-Kutta schemes}, author={Carpenter, Mark H and Kennedy, Christopher A}, year={1994} }

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OrdinaryDiffEqLowStorageRK.SHLDDRK64 — Type
SHLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            williamson_condition = true)

Explicit Runge-Kutta Method. A fourth-order, six-stage explicit low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

D. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }

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OrdinaryDiffEqLowStorageRK.SHLDDRK52 — Type
SHLDDRK52(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. TBD

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.SHLDDRK_2N — Type
SHLDDRK_2N(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. TBD

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.HSLDDRK64 — Type
HSLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            williamson_condition = true)

Explicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

D. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }

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OrdinaryDiffEqLowStorageRK.DGLDDRK73_C — Type
DGLDDRK73_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)

Explicit Runge-Kutta Method. 7-stage, third order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

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OrdinaryDiffEqLowStorageRK.DGLDDRK84_C — Type
DGLDDRK84_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)

Explicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

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OrdinaryDiffEqLowStorageRK.DGLDDRK84_F — Type
DGLDDRK84_F(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)

Explicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when the maximum spatial step size is not constrained. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

T. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024

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OrdinaryDiffEqLowStorageRK.NDBLSRK124 — Type
NDBLSRK124(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)

Explicit Runge-Kutta Method. 12-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

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OrdinaryDiffEqLowStorageRK.NDBLSRK134 — Type
NDBLSRK134(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)

Explicit Runge-Kutta Method. 13-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

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OrdinaryDiffEqLowStorageRK.NDBLSRK144 — Type
NDBLSRK144(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)

Explicit Runge-Kutta Method. 14-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)
  • williamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.

References

Jens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003

source
OrdinaryDiffEqLowStorageRK.CFRLDDRK64 — Type
CFRLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-storage, low-dissipation, low-dispersion scheme. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

M. Calvo, J. M. Franco, L. Randez. A New Minimum Storage Runge–Kutta Scheme for Computational Acoustics. Journal of Computational Physics, 201, pp 1-12, 2004. doi: https://doi.org/10.1016/j.jcp.2004.05.012

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OrdinaryDiffEqLowStorageRK.TSLDDRK74 — Type
TSLDDRK74(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 7-stage, fourth order low-storage low-dissipation, low-dispersion scheme with maximal accuracy and stability limit along the imaginary axes. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Kostas Tselios, T. E. Simos. Optimized Runge–Kutta Methods with Minimal Dispersion and Dissipation for Problems arising from Computational Acoustics. Physics Letters A, 393(1-2), pp 38-47, 2007. doi: https://doi.org/10.1016/j.physleta.2006.10.072

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OrdinaryDiffEqLowStorageRK.CKLLSRK43_2 — Type
CKLLSRK43_2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 4-stage, third order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C — Type
CKLLSRK54_3C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S — Type
CKLLSRK95_4S(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C — Type
CKLLSRK95_4C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M — Type
CKLLSRK95_4M(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R — Type
CKLLSRK54_3C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R — Type
CKLLSRK54_3M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

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OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R — Type
CKLLSRK54_3N_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

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OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R — Type
CKLLSRK85_4C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R — Type
CKLLSRK85_4M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R — Type
CKLLSRK85_4P_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R — Type
CKLLSRK54_3N_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R — Type
CKLLSRK54_3M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R — Type
CKLLSRK65_4M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. 6-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R — Type
CKLLSRK85_4FM_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                   step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R — Type
CKLLSRK75_4M_5R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. CKLLSRK754M5R: Low-Storage Method 7-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32 — Type
ParsaniKetchesonDeconinck3S32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 3-stage, second order (3S) low-storage scheme, optimized the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

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OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82 — Type
ParsaniKetchesonDeconinck3S82(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 8-stage, second order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53 — Type
ParsaniKetchesonDeconinck3S53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 5-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173 — Type
ParsaniKetchesonDeconinck3S173(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 17-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94 — Type
ParsaniKetchesonDeconinck3S94(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 9-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184 — Type
ParsaniKetchesonDeconinck3S184(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 18-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105 — Type
ParsaniKetchesonDeconinck3S105(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 10-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205 — Type
ParsaniKetchesonDeconinck3S205(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. Low-Storage Method 20-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Parsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp35 — Type
RDPK3Sp35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit Runge-Kutta method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35 — Type
RDPK3SpFSAL35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit Runge-Kutta method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp49 — Type
RDPK3Sp49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fourth-order, nine-stage explicit Runge-Kutta method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49 — Type
RDPK3SpFSAL49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fourth-order, nine-stage explicit Runge-Kutta method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3Sp510 — Type
RDPK3Sp510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fifth-order, ten-stage explicit Runge-Kutta method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source
OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510 — Type
RDPK3SpFSAL510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fifth-order, ten-stage explicit Runge-Kutta method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ranocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836

source

SSP Optimized Runge-Kutta Methods

OrdinaryDiffEqSSPRK.KYK2014DGSSPRK_3S2 — Type
KYK2014DGSSPRK_3S2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                     step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. TBD

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqSSPRK.SSPRK22 — Type
SSPRK22(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A second-order, two-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Shu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5

source
OrdinaryDiffEqSSPRK.SSPRK33 — Type
SSPRK33(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, three-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Shu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5

source
OrdinaryDiffEqSSPRK.SSPRK53 — Type
SSPRK53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.KYKSSPRK42 — Type
KYKSSPRK42(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. TBD

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

source
OrdinaryDiffEqSSPRK.SSPRK53_2N1 — Type
SSPRK53_2N1(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK53_2N2 — Type
SSPRK53_2N2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK53_H — Type
SSPRK53_H(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Higueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.

source
OrdinaryDiffEqSSPRK.SSPRK63 — Type
SSPRK63(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, six-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK73 — Type
SSPRK73(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, seven-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK83 — Type
SSPRK83(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, eight-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207

source
OrdinaryDiffEqSSPRK.SSPRK43 — Type
SSPRK43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Optimal third-order explicit SSP method with four stages discovered by

  • J. F. B. M. Kraaijevanger. "Contractivity of Runge-Kutta methods." In: BIT Numerical Mathematics 31.3 (1991), pp. 482–528. DOI: 10.1007/BF01933264.

Embedded method constructed by

  • Sidafa Conde, Imre Fekete, John N. Shadid. "Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods." arXiv: 1806.08693

Efficient implementation (and optimized controller) developed by

  • Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836
source
OrdinaryDiffEqSSPRK.SSPRK432 — Type
SSPRK432(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011. Example 6.1

source
OrdinaryDiffEqSSPRK.SSPRKMSVS43 — Type
SSPRKMSVS43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, four-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Shu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073

source
OrdinaryDiffEqSSPRK.SSPRKMSVS32 — Type
SSPRKMSVS32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A second-order, three-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Shu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073

source
OrdinaryDiffEqSSPRK.SSPRK932 — Type
SSPRK932(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A third-order, nine-stage explicit strong stability preserving (SSP) method.

Consider using SSPRK43 instead, which uses the same main method and an improved embedded method.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011.

source
OrdinaryDiffEqSSPRK.SSPRK54 — Type
SSPRK54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fourth-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ruuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207.

source
OrdinaryDiffEqSSPRK.SSPRK104 — Type
SSPRK104(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)

Explicit Runge-Kutta Method. A fourth-order, ten-stage explicit strong stability preserving (SSP) method. Fixed timestep only.

Keyword Arguments

  • stage_limiter!: function of the form limiter!(u, integrator, p, t)
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

Ketcheson, David I. Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM Journal on Scientific Computing 30.4 (2008): 2113-2136.

source
diff --git a/dev/nonstiff/nonstiff_multistep/index.html b/dev/nonstiff/nonstiff_multistep/index.html deleted file mode 100644 index dcca5de907..0000000000 --- a/dev/nonstiff/nonstiff_multistep/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Multistep Methods for Non-Stiff Equations · OrdinaryDiffEq.jl
GitHub

Multistep Methods for Non-Stiff Equations

Explicit Multistep Methods

Missing docstring.

Missing docstring for AB3. Check Documenter's build log for details.

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Missing docstring for AB4. Check Documenter's build log for details.

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Missing docstring for AB5. Check Documenter's build log for details.

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Missing docstring for AN5. Check Documenter's build log for details.

Predictor-Corrector Methods

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Missing docstring for ABM32. Check Documenter's build log for details.

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Missing docstring for ABM43. Check Documenter's build log for details.

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Missing docstring for ABM54. Check Documenter's build log for details.

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Missing docstring for VCAB3. Check Documenter's build log for details.

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Missing docstring for VCAB4. Check Documenter's build log for details.

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Missing docstring for VCAB5. Check Documenter's build log for details.

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Missing docstring for VCABM3. Check Documenter's build log for details.

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Missing docstring for VCABM4. Check Documenter's build log for details.

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Missing docstring for VCABM5. Check Documenter's build log for details.

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Missing docstring for VCABM. Check Documenter's build log for details.

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For example:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"using OrdinaryDiffEq, StaticArrays\nfunction lorenz(u, p, t)\n SA[10.0(u[2] - u[1]), u[1] * (28.0 - u[3]) - u[2], u[1] * u[2] - (8 / 3) * u[3]]\nend\nu0 = SA[1.0; 0.0; 0.0]\ntspan = (0.0, 100.0)\nprob = ODEProblem(lorenz, u0, tspan)\nsol = solve(prob, Tsit5())","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"For “refined ODEs”, like dynamical equations and SecondOrderODEProblems, refer to the DiffEqDocs. For example, in DiffEqTutorials.jl we show how to solve equations of motion using symplectic methods:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"function HH_acceleration(dv, v, u, p, t)\n x, y = u\n dx, dy = dv\n dv[1] = -x - 2x * y\n dv[2] = y^2 - y - x^2\nend\ninitial_positions = [0.0, 0.1]\ninitial_velocities = [0.5, 0.0]\nprob = SecondOrderODEProblem(HH_acceleration, initial_velocities, initial_positions, tspan)\nsol2 = solve(prob, KahanLi8(), dt = 1 / 10);","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).","category":"page"},{"location":"usage/#Available-Solvers","page":"Usage","title":"Available Solvers","text":"","category":"section"},{"location":"usage/","page":"Usage","title":"Usage","text":"For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.","category":"page"},{"location":"stiff/stabilized_rk/#Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"","category":"section"},{"location":"stiff/stabilized_rk/#Explicit-Stabilized-Runge-Kutta-Methods","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Explicit Stabilized Runge-Kutta Methods","text":"","category":"section"},{"location":"stiff/stabilized_rk/","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"Explicit stabilized methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example","category":"page"},{"location":"stiff/stabilized_rk/","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`","category":"page"},{"location":"stiff/stabilized_rk/","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"The methods ROCK2 and ROCK4 also include keyword arguments min_stages and max_stages, which specify upper and lower bounds on the adaptively chosen number of stages for stability.","category":"page"},{"location":"stiff/stabilized_rk/","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"ROCK2\nROCK4\nSERK2\nESERK4\nESERK5\nRKC","category":"page"},{"location":"stiff/stabilized_rk/#OrdinaryDiffEqStabilizedRK.ROCK2","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"OrdinaryDiffEqStabilizedRK.ROCK2","text":"Assyr Abdulle, Alexei A. Medovikov. Second Order Chebyshev Methods based on Orthogonal Polynomials. Numerische Mathematik, 90 (1), pp 1-18, 2001. doi: https://dx.doi.org/10.1007/s002110100292\n\nROCK2: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nThis method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form\n\neigen_est = (integrator) -> integrator.eigen_est = upper_bound,\n\nwhere upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/stabilized_rk/#OrdinaryDiffEqStabilizedRK.ROCK4","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"OrdinaryDiffEqStabilizedRK.ROCK4","text":"ROCK4(; min_stages = 0, max_stages = 152, eigen_est = nothing)\n\nAssyr Abdulle. Fourth Order Chebyshev Methods With Recurrence Relation. 2002 Society for Industrial and Applied Mathematics Journal on Scientific Computing, 23(6), pp 2041-2054, 2001. doi: https://doi.org/10.1137/S1064827500379549\n\nROCK4: Stabilized Explicit Method. Fourth order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nThis method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form\n\neigen_est = (integrator) -> integrator.eigen_est = upper_bound,\n\nwhere upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/stabilized_rk/#OrdinaryDiffEqStabilizedRK.ESERK4","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"OrdinaryDiffEqStabilizedRK.ESERK4","text":"ESERK4(; eigen_est = nothing)\n\nJ. Martín-Vaquero, B. Kleefeld. Extrapolated stabilized explicit Runge-Kutta methods, Journal of Computational Physics, 326, pp 141-155, 2016. doi: https://doi.org/10.1016/j.jcp.2016.08.042.\n\nESERK4: Stabilized Explicit Method. Fourth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nThis method takes the keyword argument eigen_est of the form\n\neigen_est = (integrator) -> integrator.eigen_est = upper_bound,\n\nwhere upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/stabilized_rk/#OrdinaryDiffEqStabilizedRK.ESERK5","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"OrdinaryDiffEqStabilizedRK.ESERK5","text":"ESERK5(; eigen_est = nothing)\n\nJ. Martín-Vaquero, A. Kleefeld. ESERK5: A fifth-order extrapolated stabilized explicit Runge-Kutta method, Journal of Computational and Applied Mathematics, 356, pp 22-36, 2019. doi: https://doi.org/10.1016/j.cam.2019.01.040.\n\nESERK5: Stabilized Explicit Method. Fifth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nThis method takes the keyword argument eigen_est of the form\n\neigen_est = (integrator) -> integrator.eigen_est = upper_bound,\n\nwhere upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/stabilized_rk/#OrdinaryDiffEqStabilizedRK.RKC","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"OrdinaryDiffEqStabilizedRK.RKC","text":"RKC(; eigen_est = nothing)\n\nB. P. Sommeijer, L. F. Shampine, J. G. Verwer. RKC: An Explicit Solver for Parabolic PDEs, Journal of Computational and Applied Mathematics, 88(2), pp 315-326, 1998. doi: https://doi.org/10.1016/S0377-0427(97)00219-7\n\nRKC: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues.\n\nThis method takes the keyword argument eigen_est of the form\n\neigen_est = (integrator) -> integrator.eigen_est = upper_bound,\n\nwhere upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/stabilized_rk/#Implicit-Stabilized-Runge-Kutta-Methods","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Implicit Stabilized Runge-Kutta Methods","text":"","category":"section"},{"location":"stiff/stabilized_rk/","page":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","title":"Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)","text":"IRKC","category":"page"},{"location":"dynamical/nystrom/#Runge-Kutta-Nystrom-Methods","page":"Runge-Kutta Nystrom Methods","title":"Runge-Kutta Nystrom Methods","text":"","category":"section"},{"location":"dynamical/nystrom/","page":"Runge-Kutta Nystrom Methods","title":"Runge-Kutta Nystrom Methods","text":"IRKN3\nIRKN4\nNystrom4\nNystrom4VelocityIndependent\nNystrom5VelocityIndependent\nFineRKN4\nFineRKN5\nDPRKN6\nDPRKN6FM\nDPRKN8\nDPRKN12\nERKN4\nERKN5\nERKN7","category":"page"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.IRKN3","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.IRKN3","text":"IRKN3\n\nImproved Runge-Kutta-Nyström method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only.\n\nSecond order ODE should not depend on the first derivative.\n\nReferences\n\n@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.IRKN4","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.IRKN4","text":"IRKN4\n\nImproves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only.\n\nSecond order ODE should not be dependent on the first derivative.\n\nRecommended for smooth problems with expensive functions to evaluate.\n\nReferences\n\n@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.Nystrom4","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.Nystrom4","text":"Nystrom4\n\nA 4th order explicit Runge-Kutta-Nyström method which can be applied directly on second order ODEs. Can only be used with fixed time steps.\n\nIn case the ODE Problem is not dependent on the first derivative consider using Nystrom4VelocityIndependent to increase performance.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.Nystrom4VelocityIndependent","text":"Nystrom4VelocityIdependent\n\nA 4th order explicit Runkge-Kutta-Nyström method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).\n\nMore efficient then Nystrom4 on velocity independent problems, since less evaluations are needed.\n\nFixed time steps only.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.Nystrom5VelocityIndependent","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.Nystrom5VelocityIndependent","text":"Nystrom5VelocityIndependent\n\nA 5th order explicit Runkge-Kutta-Nyström method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative). Fixed time steps only.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.FineRKN4","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.FineRKN4","text":"FineRKN4()\n\nA 4th order explicit Runge-Kutta-Nyström method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.\n\nReferences\n\n@article{fine1987low,\n title={Low order practical {R}unge-{K}utta-{N}ystr{\"o}m methods},\n author={Fine, Jerry Michael},\n journal={Computing},\n volume={38},\n number={4},\n pages={281--297},\n year={1987},\n publisher={Springer}\n}\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.FineRKN5","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.FineRKN5","text":"FineRKN5()\n\nA 5th order explicit Runge-Kutta-Nyström method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.\n\nReferences\n\n@article{fine1987low,\n title={Low order practical {R}unge-{K}utta-{N}ystr{\"o}m methods},\n author={Fine, Jerry Michael},\n journal={Computing},\n volume={38},\n number={4},\n pages={281--297},\n year={1987},\n publisher={Springer}\n}\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.DPRKN6","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.DPRKN6","text":"DPRKN6\n\n6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative. Free 6th order interpolant.\n\nReferences\n\n@article{dormand1987runge, title={Runge-kutta-nystrom triples}, author={Dormand, JR and Prince, PJ}, journal={Computers \\& Mathematics with Applications}, volume={13}, number={12}, pages={937–949}, year={1987}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.DPRKN6FM","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.DPRKN6FM","text":"DPRKN6FM\n\n6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.\n\nCompared to DPRKN6, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.\n\nReferences\n\n@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.DPRKN8","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.DPRKN8","text":"DPRKN8\n\n8th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.\n\nNot as efficient as DPRKN12 when high accuracy is needed, however this solver is competitive with DPRKN6 at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.\n\nReferences\n\n@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.DPRKN12","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.DPRKN12","text":"DPRKN12\n\n12th order explicit Rugne-Kutta-Nyström method. The second order ODE should not depend on the first derivative.\n\nMost efficient when high accuracy is needed.\n\nReferences\n\n@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.ERKN4","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.ERKN4","text":"ERKN4\n\nEmbedded 4(3) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.\n\nThe second order ODE should not depend on the first derivative.\n\nUses adaptive step size control. This method is extra efficient on periodic problems.\n\nReferences\n\n@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{\"o}m Method for Solving Periodic Initial Value Problems}, author={Demba, MA and Senu, N and Ismail, F}, journal={Applied Mathematical Sciences}, volume={11}, number={17}, pages={819–838}, year={2017} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.ERKN5","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.ERKN5","text":"ERKN5\n\nEmbedded 5(4) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.\n\nThe second order ODE should not depend on the first derivative.\n\nUses adaptive step size control. This method is extra efficient on periodic problems.\n\nReferences\n\n@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{\"o}m Methods for the Numerical Solution of Oscillatory Initial Value Problems}, author={Demba, Musa A and Senu, Norazak and Ismail, Fudziah}, journal={Mathematical and Computational Applications}, volume={21}, number={4}, pages={46}, year={2016}, publisher={Multidisciplinary Digital Publishing Institute} }\n\n\n\n\n\n","category":"type"},{"location":"dynamical/nystrom/#OrdinaryDiffEqRKN.ERKN7","page":"Runge-Kutta Nystrom Methods","title":"OrdinaryDiffEqRKN.ERKN7","text":"ERKN7\n\nEmbedded pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.\n\nThe second order ODE should not depend on the first derivative.\n\nUses adaptive step size control. This method is extra efficient on periodic Problems.\n\nReferences\n\n@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{\"o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753} }\n\n\n\n\n\n","category":"type"},{"location":"stiff/implicit_extrapolation/#Implicit-Extrapolation-Methods","page":"Implicit Extrapolation Methods","title":"Implicit Extrapolation Methods","text":"","category":"section"},{"location":"stiff/implicit_extrapolation/","page":"Implicit Extrapolation Methods","title":"Implicit Extrapolation Methods","text":"ImplicitEulerExtrapolation\nImplicitDeuflhardExtrapolation\nImplicitHairerWannerExtrapolation\nImplicitEulerBarycentricExtrapolation","category":"page"},{"location":"stiff/implicit_extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitEulerExtrapolation","page":"Implicit Extrapolation Methods","title":"OrdinaryDiffEqExtrapolation.ImplicitEulerExtrapolation","text":"ImplicitEulerExtrapolation: Parallelized Implicit Extrapolation Method Extrapolation of implicit Euler method with Romberg sequence. Similar to Hairer's SEULEX.\n\n\n\n\n\n","category":"type"},{"location":"stiff/implicit_extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation","page":"Implicit Extrapolation Methods","title":"OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation","text":"ImplicitDeuflhardExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates\n\n\n\n\n\n","category":"type"},{"location":"stiff/implicit_extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation","page":"Implicit Extrapolation Methods","title":"OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation","text":"ImplicitHairerWannerExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.\n\n\n\n\n\n","category":"type"},{"location":"stiff/implicit_extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation","page":"Implicit Extrapolation Methods","title":"OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation","text":"ImplicitEulerBarycentricExtrapolation: Parallelized Implicit Extrapolation Method Euler extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#IMEX-SDIRK-Methods","page":"IMEX SDIRK Methods","title":"IMEX SDIRK Methods","text":"","category":"section"},{"location":"imex/imex_sdirk/","page":"IMEX SDIRK Methods","title":"IMEX SDIRK Methods","text":"IMEXEuler\nIMEXEulerARK\nKenCarp3\nKenCarp4\nKenCarp47\nKenCarp5\nKenCarp58\nESDIRK54I8L2SA\nESDIRK436L2SA2\nESDIRK437L2SA\nESDIRK547L2SA2\nESDIRK659L2SA","category":"page"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqBDF.IMEXEuler","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqBDF.IMEXEuler","text":"IMEXEuler(;kwargs...)\n\nThe one-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nWhen applied to a SplitODEProblem of the form\n\nu'(t) = f1(u) + f2(u)\n\nThe default IMEXEuler() method uses an update of the form\n\nunew = uold + dt * (f1(unew) + f2(uold))\n\nSee also SBDF, IMEXEulerARK.\n\n\n\n\n\n","category":"function"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqBDF.IMEXEulerARK","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqBDF.IMEXEulerARK","text":"IMEXEulerARK(;kwargs...)\n\nThe one-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nWhen applied to a SplitODEProblem of the form\n\nu'(t) = f1(u) + f2(u)\n\nA classical additive Runge-Kutta method in the sense of Araújo, Murua, Sanz-Serna (1997) consisting of the implicit and the explicit Euler method given by\n\ny1 = uold + dt * f1(y1)\nunew = uold + dt * (f1(unew) + f2(y1))\n\nSee also SBDF, IMEXEuler.\n\n\n\n\n\n","category":"function"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.KenCarp3","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.KenCarp3","text":"@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }\n\nKenCarp3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.KenCarp4","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.KenCarp4","text":"@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }\n\nKenCarp4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method with splitting\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.KenCarp47","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.KenCarp47","text":"@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier} }\n\nKenCarp47: SDIRK Method An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.KenCarp5","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.KenCarp5","text":"@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center} }\n\nKenCarp5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method with splitting\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.KenCarp58","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.KenCarp58","text":"@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier} }\n\nKenCarp58: SDIRK Method An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.ESDIRK436L2SA2","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.ESDIRK436L2SA2","text":"@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.ESDIRK437L2SA","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.ESDIRK437L2SA","text":"@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.ESDIRK547L2SA2","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.ESDIRK547L2SA2","text":"@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"imex/imex_sdirk/#OrdinaryDiffEqSDIRK.ESDIRK659L2SA","page":"IMEX SDIRK Methods","title":"OrdinaryDiffEqSDIRK.ESDIRK659L2SA","text":"@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244}\n\nCurrently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details. }\n\n\n\n\n\n","category":"type"},{"location":"#OrdinaryDiffEq.jl","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.","category":"page"},{"location":"#Installation","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"Installation","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"import Pkg;\nPkg.add(\"OrdinaryDiffEq\");","category":"page"},{"location":"#Reproducibility","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"You can also download the \nmanifest file and the\nproject file.","category":"page"},{"location":"stiff/sdirk/#Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","text":"","category":"section"},{"location":"stiff/sdirk/","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","text":"ImplicitEuler\nImplicitMidpoint\nTrapezoid\nTRBDF2\nSDIRK2\nSDIRK22\nSSPSDIRK2\nKvaerno3\nCFNLIRK3\nCash4\nSFSDIRK4\nSFSDIRK5\nSFSDIRK6\nSFSDIRK7\nSFSDIRK8\nHairer4\nHairer42\nKvaerno4\nKvaerno5","category":"page"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.ImplicitEuler","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.ImplicitEuler","text":"ImplicitEuler: SDIRK Method A 1st order implicit solver. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.ImplicitMidpoint","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.ImplicitMidpoint","text":"ImplicitMidpoint: SDIRK Method A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Trapezoid","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Trapezoid","text":"Andre Vladimirescu. 1994. The Spice Book. John Wiley & Sons, Inc., New York, NY, USA.\n\nTrapezoid: SDIRK Method A second order A-stable symmetric ESDIRK method. \"Almost symplectic\" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences approximation to the second derivative terms in the local truncation error estimate (the SPICE approximation strategy).\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.TRBDF2","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.TRBDF2","text":"@article{hosea1996analysis, title={Analysis and implementation of TR-BDF2}, author={Hosea, ME and Shampine, LF}, journal={Applied Numerical Mathematics}, volume={20}, number={1-2}, pages={21–37}, year={1996}, publisher={Elsevier} }\n\nTRBDF2: SDIRK Method A second order A-B-L-S-stable one-step ESDIRK method. Includes stiffness-robust error estimates for accurate adaptive timestepping, smoothed derivatives for highly stiff and oscillatory problems.\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.SDIRK2","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.SDIRK2","text":"@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM} }\n\nSDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Kvaerno3","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Kvaerno3","text":"@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }\n\nKvaerno3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Cash4","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Cash4","text":"@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM} }\n\nCash4: SDIRK Method An A-L stable 4th order SDIRK method\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Hairer4","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Hairer4","text":"E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\nHairer4: SDIRK Method An A-L stable 4th order SDIRK method\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Hairer42","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Hairer42","text":"E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\nHairer42: SDIRK Method An A-L stable 4th order SDIRK method\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Kvaerno4","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Kvaerno4","text":"@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }\n\nKvaerno4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method.\n\n\n\n\n\n","category":"type"},{"location":"stiff/sdirk/#OrdinaryDiffEqSDIRK.Kvaerno5","page":"Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods","title":"OrdinaryDiffEqSDIRK.Kvaerno5","text":"@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }\n\nKvaerno5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method\n\n\n\n\n\n","category":"type"}] +[{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"","category":"section"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"A symplectic integrator is an integrator whose solution resides on a symplectic manifold. Because of discretization error, when it is solving a Hamiltonian system it doesn't get exactly the correct trajectory on the manifold. Instead, that trajectory itself is perturbed O(Δtn) for the order n from the true trajectory. Then there's a linear drift due to numerical error of this trajectory over time Normal integrators tend to have a quadratic (or more) drift, and do not have any good global guarantees about this phase space path (just local). What means is that symplectic integrators tend to capture the long-time patterns better than normal integrators because of this lack of drift and this almost guarantee of periodicity.","category":"page"},{"location":"explicit/SymplecticRK/#Installation","page":"OrdinaryDiffEqSymplecticRK","title":"Installation","text":"","category":"section"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"To be able to access the solvers in OrdinaryDiffEqSymplecticRK, you must first install them use the Julia package manager:","category":"page"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqSymplecticRK\")","category":"page"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"explicit/SymplecticRK/#Example-usage","page":"OrdinaryDiffEqSymplecticRK","title":"Example usage","text":"","category":"section"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"using OrdinaryDiffEqSymplecticRK\nfunction HH_acceleration!(dv, v, u, p, t)\n x, y = u\n dx, dy = dv\n dv[1] = -x - 2x * y\n dv[2] = y^2 - y - x^2\nend\ninitial_positions = [0.0, 0.1]\ninitial_velocities = [0.5, 0.0]\ntspan = (0.0, 1.0)\nprob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)\nsol = solve(prob, KahanLi8(), dt = 1 / 10)","category":"page"},{"location":"explicit/SymplecticRK/#Full-list-of-solvers","page":"OrdinaryDiffEqSymplecticRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/SymplecticRK/","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK","text":"SymplecticEuler\nVelocityVerlet\nVerletLeapfrog\nPseudoVerletLeapfrog\nMcAte2\nRuth3\nMcAte3\nCandyRoz4\nMcAte4\nCalvoSanz4\nMcAte42\nMcAte5\nYoshida6\nKahanLi6\nMcAte8\nKahanLi8\nSofSpa10","category":"page"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.SymplecticEuler","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.SymplecticEuler","text":"SymplecticEuler()\n\nSymplectic Runge-Kutta Methods First order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\nhttps://en.wikipedia.org/wiki/Semi-implicitEulermethod\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.VelocityVerlet","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.VelocityVerlet","text":"VelocityVerlet()\n\nSymplectic Runge-Kutta Methods 2nd order explicit symplectic integrator. Requires f_2(t,u) = v, i.e. a second order ODE.\n\nKeyword Arguments\n\nReferences\n\n@article{verlet1967computer, title={Computer\" experiments\" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.VerletLeapfrog","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.VerletLeapfrog","text":"VerletLeapfrog()\n\nSymplectic Runge-Kutta Methods 2nd order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{verlet1967computer, title={Computer\" experiments\" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.PseudoVerletLeapfrog","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.PseudoVerletLeapfrog","text":"PseudoVerletLeapfrog()\n\nSymplectic Runge-Kutta Methods 2nd order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{verlet1967computer, title={Computer\" experiments\" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, author={Verlet, Loup}, journal={Physical review}, volume={159}, number={1}, pages={98}, year={1967}, publisher={APS} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte2","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte2","text":"McAte2()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 2nd order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.Ruth3","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.Ruth3","text":"Ruth3()\n\nSymplectic Runge-Kutta Methods 3rd order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{ruth1983canonical, title={A canonical integration technique}, author={Ruth, Ronald D}, journal={IEEE Trans. Nucl. Sci.}, volume={30}, number={CERN-LEP-TH-83-14}, pages={2669–2671}, year={1983}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte3","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte3","text":"McAte3()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 3rd order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.CandyRoz4","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.CandyRoz4","text":"CandyRoz4()\n\nSymplectic Runge-Kutta Methods 4th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{candy1991symplectic, itle={A symplectic integration algorithm for separable Hamiltonian functions}, uthor={Candy, J and Rozmus, W}, ournal={Journal of Computational Physics}, olume={92}, umber={1}, ages={230–256}, ear={1991}, ublisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte4","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte4","text":"McAte4()\n\nSymplectic Runge-Kutta Methods 4th order explicit symplectic integrator. Requires quadratic kinetic energy.\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.CalvoSanz4","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.CalvoSanz4","text":"CalvoSanz4()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 4th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{sanz1993symplectic, title={Symplectic numerical methods for Hamiltonian problems}, author={Sanz-Serna, Jes{'u}s Maria and Calvo, Mari-Paz}, journal={International Journal of Modern Physics C}, volume={4}, number={02}, pages={385–392}, year={1993}, publisher={World Scientific} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte42","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte42","text":"McAte42()\n\nSymplectic Runge-Kutta Methods 4th order explicit symplectic integrator. BROKEN\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte5","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte5","text":"McAte5()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 5th order explicit symplectic integrator. Requires quadratic kinetic energy.\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1992accuracy, title={The accuracy of symplectic integrators}, author={McLachlan, Robert I and Atela, Pau}, journal={Nonlinearity}, volume={5}, number={2}, pages={541}, year={1992}, publisher={IOP Publishing} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.Yoshida6","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.Yoshida6","text":"Yoshida6()\n\nSymplectic Runge-Kutta Methods 6th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{yoshida1990construction, title={Construction of higher order symplectic integrators}, author={Yoshida, Haruo}, journal={Physics letters A}, volume={150}, number={5-7}, pages={262–268}, year={1990}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.KahanLi6","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.KahanLi6","text":"KahanLi6()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 6th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{yoshida1990construction, title={Construction of higher order symplectic integrators}, author={Yoshida, Haruo}, journal={Physics letters A}, volume={150}, number={5-7}, pages={262–268}, year={1990}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.McAte8","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.McAte8","text":"McAte8()\n\nSymplectic Runge-Kutta Methods 8th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{mclachlan1995numerical, title={On the numerical integration of ordinary differential equations by symmetric composition methods}, author={McLachlan, Robert I}, journal={SIAM Journal on Scientific Computing}, volume={16}, number={1}, pages={151–168}, year={1995}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.KahanLi8","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.KahanLi8","text":"KahanLi8()\n\nSymplectic Runge-Kutta Methods Optimized efficiency 8th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{kahan1997composition, title={Composition constants for raising the orders of unconventional schemes for ordinary differential equations}, author={Kahan, William and Li, Ren-Cang}, journal={Mathematics of computation}, volume={66}, number={219}, pages={1089–1099}, year={1997}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SymplecticRK/#OrdinaryDiffEqSymplecticRK.SofSpa10","page":"OrdinaryDiffEqSymplecticRK","title":"OrdinaryDiffEqSymplecticRK.SofSpa10","text":"SofSpa10()\n\nSymplectic Runge-Kutta Methods 10th order explicit symplectic integrator.\n\nKeyword Arguments\n\nReferences\n\n@article{sofroniou2005derivation, title={Derivation of symmetric composition constants for symmetric integrators}, author={Sofroniou, Mark and Spaletta, Giulia}, journal={Optimization Methods and Software}, volume={20}, number={4-5}, pages={597–613}, year={2005}, publisher={Taylor \\& Francis}}\n\n\n\n\n\n","category":"type"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"CollapsedDocStrings = true","category":"page"},{"location":"imex/IMEXMultistep/#OrdinaryDiffEqIMEXMultistep","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"","category":"section"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"Solvers if your system of ordinary differential equations can be split up into the sum of a stiff and non-stiff part.","category":"page"},{"location":"imex/IMEXMultistep/#Installation","page":"OrdinaryDiffEqIMEXMultistep","title":"Installation","text":"","category":"section"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"To be able to access the solvers in OrdinaryDiffEqIMEXMultistep, you must first install them use the Julia package manager:","category":"page"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqIMEXMultistep\")","category":"page"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"imex/IMEXMultistep/#Example-usage","page":"OrdinaryDiffEqIMEXMultistep","title":"Example usage","text":"","category":"section"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"using OrdinaryDiffEqIMEXMultistep\nf1 = (u, p, t) -> 2u\nf2 = (u, p, t) -> 2u\nu0 = 1.0\ntspan = (0.0, 1.0)\nprob = SplitODEProblem(f1, f2, u0, tspan)\nsol = solve(prob,CNAB2(), dt = 1 / 10)","category":"page"},{"location":"imex/IMEXMultistep/#Full-list-of-solvers","page":"OrdinaryDiffEqIMEXMultistep","title":"Full list of solvers","text":"","category":"section"},{"location":"imex/IMEXMultistep/","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep","text":"CNAB2\nCNLF2","category":"page"},{"location":"imex/IMEXMultistep/#OrdinaryDiffEqIMEXMultistep.CNAB2","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep.CNAB2","text":"CNAB2()\n\nIMEX Multistep method. Crank-Nicholson Adams-Bashforth 2.\n\nKeyword Arguments\n\nReferences\n\n@article{jorgenson2014unconditional, title={Unconditional stability of a Crank-Nicolson Adams-Bashforth 2 numerical method}, author={JORGENSON, ANDREW D}, journal={A (A- C)}, volume={1}, number={2}, pages={1}, year={2014}} @article{he2010numerical, title={Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time-dependent Navier–Stokes equations}, author={He, Yinnian and Li, Jian}, journal={International journal for numerical methods in fluids}, volume={62}, number={6}, pages={647–659}, year={2010}, publisher={Wiley Online Library}}\n\n\n\n\n\n","category":"type"},{"location":"imex/IMEXMultistep/#OrdinaryDiffEqIMEXMultistep.CNLF2","page":"OrdinaryDiffEqIMEXMultistep","title":"OrdinaryDiffEqIMEXMultistep.CNLF2","text":"CNLF2()\n\nIMEX Multistep method. Crank-Nicholson Leapfrong 2.\n\nKeyword Arguments\n\nReferences\n\n@article{han2020second, title={A second order, linear, unconditionally stable, Crank–Nicolson–Leapfrog scheme for phase field models of two-phase incompressible flows}, author={Han, Daozhi and Jiang, Nan}, journal={Applied Mathematics Letters}, volume={108}, pages={106521}, year={2020}, publisher={Elsevier}} @article{jiang2015crank, title={A Crank–Nicolson Leapfrog stabilization: Unconditional stability and two applications}, author={Jiang, Nan and Kubacki, Michaela and Layton, William and Moraiti, Marina and Tran, Hoang}, journal={Journal of Computational and Applied Mathematics}, volume={281}, pages={263–276}, year={2015}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"CollapsedDocStrings = true","category":"page"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"Methods for semi-linear differential equations.","category":"page"},{"location":"semilinear/Linear/#Installation","page":"OrdinaryDiffEqLinear","title":"Installation","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"To be able to access the solvers in OrdinaryDiffEqLinear, you must first install them use the Julia package manager:","category":"page"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqLinear\")","category":"page"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"semilinear/Linear/#Example-usage","page":"OrdinaryDiffEqLinear","title":"Example usage","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"using OrdinaryDiffEqLinear, SciMLOperators\nfunction update_func(A, u, p, t)\n A[1, 1] = 0\n A[2, 1] = sin(u[1])\n A[1, 2] = -1\n A[2, 2] = 0\nend\nA0 = ones(2, 2)\nA = DiffEqArrayOperator(A0, update_func = update_func)\nu0 = ones(2)\ntspan = (0.0, 30.0)\nprob = ODEProblem(A, u0, tspan)\nsol = solve(prob, LieRK4(), dt = 1 / 4)","category":"page"},{"location":"semilinear/Linear/#Full-list-of-solvers","page":"OrdinaryDiffEqLinear","title":"Full list of solvers","text":"","category":"section"},{"location":"semilinear/Linear/#Time-and-State-Independent-Solvers","page":"OrdinaryDiffEqLinear","title":"Time and State-Independent Solvers","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"LinearExponential","category":"page"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.LinearExponential","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.LinearExponential","text":"LinearExponential(; krylov = :off,\n m = 10,\n iop = 0)\n\nSemilinear ODE solver Exact solution formula for linear, time-independent problems.\n\nKeyword Arguments\n\nkrylov:\n:off: cache the operator beforehand. Requires Matrix(A) method defined for the operator A.\n:simple: uses simple Krylov approximations with fixed subspace size m.\n:adaptive: uses adaptive Krylov approximations with internal timestepping.\nm: Controls the size of Krylov subspace if krylov=:simple, and the initial subspace size if krylov=:adaptive.\niop: If not zero, determines the length of the incomplete orthogonalization procedure Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@book{strogatz2018nonlinear, title={Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering}, author={Strogatz, Steven H}, year={2018}, publisher={CRC press}}\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#Time-Dependent-and-State-Independent-Solvers","page":"OrdinaryDiffEqLinear","title":"Time-Dependent and State-Independent Solvers","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"MagnusMidpoint\nMagnusLeapfrog\nMagnusGauss4\nMagnusNC6\nMagnusGL6\nMagnusGL8\nMagnusNC8\nMagnusGL4","category":"page"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusMidpoint","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusMidpoint","text":"MagnusMidpoint(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Second order Magnus Midpoint method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nhttps://joshuagoings.com/2017/06/15/magnus/\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusLeapfrog","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusLeapfrog","text":"MagnusLeapfrog(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Second order Magnus Leapfrog method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nhttps://joshuagoings.com/2017/06/15/magnus/\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusGauss4","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusGauss4","text":"MagnusGauss4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Fourth order Magnus method approximated using a two stage Gauss quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{hairer2011solving, title={Solving differential equations on manifolds}, author={Hairer, Ernst}, journal={Lecture notes}, year={2011} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusNC6","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusNC6","text":"MagnusNC6(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Sixth order Magnus method approximated using Newton-Cotes quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusGL6","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusGL6","text":"MagnusGL6(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Sixth order Magnus method approximated using Gauss-Legendre quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusGL8","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusGL8","text":"MagnusGL8(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Eighth order Magnus method approximated using Newton-Cotes quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusNC8","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusNC8","text":"MagnusNC8(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Eighth order Magnus method approximated using Gauss-Legendre quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusGL4","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusGL4","text":"MagnusGL4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Fourth order Magnus method approximated using Gauss-Legendre quadrature.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{blanes2009magnus, title={The Magnus expansion and some of its applications}, author={Blanes, Sergio and Casas, Fernando and Oteo, Jose-Angel and Ros, Jos{'e}}, journal={Physics reports}, volume={470}, number={5-6}, pages={151–238}, year={2009}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#State-Dependent-Solvers","page":"OrdinaryDiffEqLinear","title":"State-Dependent Solvers","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"LieEuler\nRKMK2\nRKMK4\nLieRK4\nCG2\nCG4a\nMagnusAdapt4\nCayleyEuler","category":"page"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.LieEuler","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.LieEuler","text":"LieEuler(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver description\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{\u0007a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.RKMK2","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.RKMK2","text":"RKMK2(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Second order Runge–Kutta–Munthe-Kaas method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{\u0007a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.RKMK4","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.RKMK4","text":"RKMK4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Fourth order Runge–Kutta–Munthe-Kaas method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{\u0007a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.LieRK4","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.LieRK4","text":"LieRK4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Fourth order Lie Runge-Kutta method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{\u0007a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.CG2","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.CG2","text":"CG2(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Second order Crouch–Grossman method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{\u0007a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.CG4a","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.CG4a","text":"CG4a(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Fourth order Crouch-Grossman method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{jackiewicz2000construction, title={Construction of Runge–Kutta methods of Crouch–Grossman type of high order}, author={Jackiewicz, Zdzislaw and Marthinsen, Arne and Owren, Brynjulf}, journal={Advances in Computational Mathematics}, volume={13}, pages={405–415}, year={2000}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.MagnusAdapt4","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.MagnusAdapt4","text":"MagnusAdapt4()\n\nSemilinear ODE solver Fourth Order Adaptive Magnus method.\n\nKeyword Arguments\n\nReferences\n\n@article{li2008adaptive, title={Adaptive explicit Magnus numerical method for nonlinear dynamical systems}, author={Li, Wen-cheng and Deng, Zi-chen}, journal={Applied Mathematics and Mechanics}, volume={29}, number={9}, pages={1111–1118}, year={2008}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.CayleyEuler","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.CayleyEuler","text":"CayleyEuler()\n\nSemilinear ODE solver First order method using Cayley transformations.\n\nKeyword Arguments\n\nReferences\n\n@article{iserles2000lie, title={Lie-group methods}, author={Iserles, Arieh and Munthe-Kaas, Hans Z and Norsett, Syvert P and Zanna, Antonella}, journal={Acta numerica}, volume={9}, pages={215–365}, year={2000}, publisher={Cambridge University Press}}\n\n\n\n\n\n","category":"type"},{"location":"semilinear/Linear/#Time-and-State-Dependent-Operators","page":"OrdinaryDiffEqLinear","title":"Time and State-Dependent Operators","text":"","category":"section"},{"location":"semilinear/Linear/","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear","text":"CG3","category":"page"},{"location":"semilinear/Linear/#OrdinaryDiffEqLinear.CG3","page":"OrdinaryDiffEqLinear","title":"OrdinaryDiffEqLinear.CG3","text":"CG3(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver Third order Crouch-Grossman method.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\n@article{crouch1993numerical, title={Numerical integration of ordinary differential equations on manifolds}, author={Crouch, Peter E and Grossman, R}, journal={Journal of Nonlinear Science}, volume={3}, pages={1–33}, year={1993}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/PDIRK/","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/PDIRK/#OrdinaryDiffEqPDIRK","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"","category":"section"},{"location":"implicit/PDIRK/","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"PDIRK methods are parallel DIRK methods. SDIRK methods, or singly-diagonally implicit methods, have to build and solve a factorize a Jacobian of the form W = I-gammaJ where gamma is dependent on the chosen method. PDIRK methods use multiple different choices of gamma, i.e. W_i = I-gamma_iJ, which are all used in the update process. There are some advantages to this, as no SDIRK method can be a higher order than 5, while DIRK methods generally can have arbitrarily high order and lower error coefficients, leading to lower errors at larger dt sizes. With the right construction of the tableau, these matrices can be factorized and the underlying steps can be computed in parallel, which is why these are the parallel DIRK methods.","category":"page"},{"location":"implicit/PDIRK/","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"warning: Experimental\nOrdinaryDiffEqPDIRK is experimental, as there are no parallel DIRK tableaus that achieve good performance in the literature.","category":"page"},{"location":"implicit/PDIRK/","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqPDIRK\", \"PDIRK44\")","category":"page"},{"location":"implicit/PDIRK/#Full-list-of-solvers","page":"OrdinaryDiffEqPDIRK","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/PDIRK/","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK","text":"PDIRK44","category":"page"},{"location":"implicit/PDIRK/#OrdinaryDiffEqPDIRK.PDIRK44","page":"OrdinaryDiffEqPDIRK","title":"OrdinaryDiffEqPDIRK.PDIRK44","text":"PDIRK44(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :constant,\n thread = OrdinaryDiffEq.True())\n\nParallel Diagonally Implicit Runge-Kutta Method. A 2 processor 4th order diagonally non-adaptive implicit method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify PDIRK44(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD,\nextrapolant: TBD,\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n\"@article{iserles1990theory, title={On the theory of parallel Runge—Kutta methods}, author={Iserles, Arieh and Norrsett, SP}, journal={IMA Journal of numerical Analysis}, volume={10}, number={4}, pages={463–488}, year={1990}, publisher={Oxford University Press}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/Extrapolation/#StiffExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"","category":"section"},{"location":"implicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"Solvers based on within method parallelism. These solvers perform well for medium sized systems of ordinary differential equations, of about 20 to 500 equations, at low tolerances.","category":"page"},{"location":"implicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqExtrapolation\", \"ImplicitEulerBarycentricExtrapolation\")","category":"page"},{"location":"implicit/Extrapolation/#Full-list-of-solvers","page":"OrdinaryDiffEqExtrapolation","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"ImplicitEulerExtrapolation\nImplicitDeuflhardExtrapolation\nImplicitHairerWannerExtrapolation\nImplicitEulerBarycentricExtrapolation","category":"page"},{"location":"implicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitEulerExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ImplicitEulerExtrapolation","text":"ImplicitEulerExtrapolation(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n max_order = 12,\n min_order = 3,\n init_order = 5,\n thread = OrdinaryDiffEq.False(),\n sequence = :harmonic)\n\nParallelized Explicit Extrapolation Method. Extrapolation of implicit Euler method with Romberg sequence. Similar to Hairer's SEULEX.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEulerExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation","text":"ImplicitDeuflhardExtrapolation(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n max_order = 10,\n min_order = 1,\n init_order = 5,\n thread = OrdinaryDiffEq.False(),\n sequence = :harmonic)\n\nParallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitDeuflhardExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation","text":"ImplicitHairerWannerExtrapolation(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n max_order = 10,\n min_order = 2,\n init_order = 5,\n thread = OrdinaryDiffEq.False(),\n sequence = :harmonic)\n\nParallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitHairerWannerExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation","text":"ImplicitEulerBarycentricExtrapolation(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n max_order = 10,\n min_order = 3,\n init_order = 5,\n thread = OrdinaryDiffEq.False(),\n sequence = :harmonic,\n sequence_factor = 2)\n\nParallelized Explicit Extrapolation Method. Euler extrapolation using Barycentric coordinates, following Hairer's SODEX in the adaptivity behavior.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEulerBarycentricExtrapolation(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\nsequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/FIRK/#OrdinaryDiffEqFIRK","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"","category":"section"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"FIRK methods are fully implicit Runge-Kutta methods. They can have special properties, like be symplectic integrators, and can achieve higher order for the same number of stage in comparison to diagonal methods. However, the fully implicit methods have a larger implicit system to solve and thus have a higher linear algebra cost. This can be useful in some contexts to promote more parallelism, but also since the size of the factorization is cubic and the dominant cost for large equations, multiplying O(n^3) operations to O((sn)^3) can be a considerable cost increase for FIRK tableaus, where s, the number of stages, is particularly large. That said, the restriction to diagonal implicitness imposes order restrictions, such as SDIRK methods having a maximum order of 5, which can restrict the problems best suited for SDIRK methods.","category":"page"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"The most common FIRK method in production are those based on RadauIIA tableaus, which is an ODE representation of Gaussian collocation. Like Gaussian collocation, it achieves higher order convergence than its stages, namely order 2s+1 for s stages. Thus RadauIIA FIRK methods tend to be some of the highest order methods (excluding extrapolation methods). This means that high order RadauIIA methods are recommended in the same scenarios that high-order explicit Runge-Kutta methods are recommended simply with the restriction of being a stiff equation. Such scenarios include cases like very low tolerances: RadauIIA methods can be the best performing methods for scenarios where tolerances are 1e-9 and below. Additionally, for ODE systems of size less than 200, the increased size of the Jacobian is mitigated by improved multithreading, since BLAS implementations are only good at multithreading LU factorizations after a certain matrix size. For this reason, RadauIIA methods tend to be recommended in cases where ODE size is small to intermediate and very accurate solutions are required.","category":"page"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"They should be tested against the parallel implicit extrapolation which also specialize in this regime.","category":"page"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqFIRK\", \"RadauIIA5\")","category":"page"},{"location":"implicit/FIRK/#Full-list-of-solvers","page":"OrdinaryDiffEqFIRK","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/FIRK/","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK","text":"RadauIIA3\nRadauIIA5\nRadauIIA9","category":"page"},{"location":"implicit/FIRK/#OrdinaryDiffEqFIRK.RadauIIA3","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK.RadauIIA3","text":"RadauIIA3(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n extrapolant = :dense,\n smooth_est = true,\n step_limiter! = trivial_limiter!)\n\nFully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nextrapolant: TBD\nsmooth_est: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/FIRK/#OrdinaryDiffEqFIRK.RadauIIA5","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK.RadauIIA5","text":"RadauIIA5(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n extrapolant = :dense,\n smooth_est = true,\n step_limiter! = trivial_limiter!)\n\nFully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nextrapolant: TBD\nsmooth_est: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/FIRK/#OrdinaryDiffEqFIRK.RadauIIA9","page":"OrdinaryDiffEqFIRK","title":"OrdinaryDiffEqFIRK.RadauIIA9","text":"RadauIIA9(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n extrapolant = :dense,\n smooth_est = true,\n step_limiter! = trivial_limiter!)\n\nFully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA9(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nextrapolant: TBD\nsmooth_est: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabalizedRK","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"","category":"section"},{"location":"implicit/StabalizedRK/","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"Explicit stabilized methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example","category":"page"},{"location":"implicit/StabalizedRK/","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`","category":"page"},{"location":"implicit/StabalizedRK/","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqStabalizedRK\", \"ROCK4\")","category":"page"},{"location":"implicit/StabalizedRK/#Full-list-of-solvers","page":"OrdinaryDiffEqStabalizedRK","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/StabalizedRK/","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabalizedRK","text":"ROCK2 \nROCK4 \nRKC\nSERK2\nESERK4\nESERK5","category":"page"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.ROCK2","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.ROCK2","text":"ROCK2(; min_stages = 0,\n max_stages = 200,\n eigen_est = nothing)\n\nStabilized Explicit Method. Second order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nKeyword Arguments\n\nmin_stages: The minimum degree of the Chebyshev polynomial.\nmax_stages: The maximumdegree of the Chebyshev polynomial.\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nAssyr Abdulle, Alexei A. Medovikov. Second Order Chebyshev Methods based on Orthogonal Polynomials. Numerische Mathematik, 90 (1), pp 1-18, 2001. doi: https://dx.doi.org/10.1007/s002110100292\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.ROCK4","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.ROCK4","text":"ROCK4(; min_stages = 0,\n max_stages = 152,\n eigen_est = nothing)\n\nStabilized Explicit Method. Fourth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nKeyword Arguments\n\nmin_stages: The minimum degree of the Chebyshev polynomial.\nmax_stages: The maximumdegree of the Chebyshev polynomial.\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nAssyr Abdulle. Fourth Order Chebyshev Methods With Recurrence Relation. 2002 Society for Industrial and Applied Mathematics Journal on Scientific Computing, 23(6), pp 2041-2054, 2001. doi: https://doi.org/10.1137/S1064827500379549\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.RKC","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.RKC","text":"RKC(; eigen_est = nothing)\n\nStabilized Explicit Method. Second order method. Exhibits high stability for real eigenvalues.\n\nKeyword Arguments\n\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nB. P. Sommeijer, L. F. Shampine, J. G. Verwer. RKC: An Explicit Solver for Parabolic PDEs, Journal of Computational and Applied Mathematics, 88(2), pp 315-326, 1998. doi: https://doi.org/10.1016/S0377-0427(97)00219-7\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.SERK2","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.SERK2","text":"SERK2(; controller = :PI\n eigen_est = nothing)\n\nStabilized Explicit Method. Second order method.\n\nKeyword Arguments\n\ncontroller: TBD\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\n@article{kleefeld2013serk2v2, title={SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems}, author={Kleefeld, B and Martin-Vaquero, J}, journal={Numerical Methods for Partial Differential Equations}, volume={29}, number={1}, pages={170–185}, year={2013}, publisher={Wiley Online Library}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.ESERK4","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.ESERK4","text":"ESERK4(; eigen_est = nothing)\n\nStabilized Explicit Method. Fourth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nKeyword Arguments\n\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nJ. Martín-Vaquero, B. Kleefeld. Extrapolated stabilized explicit Runge-Kutta methods, Journal of Computational Physics, 326, pp 141-155, 2016. doi: https://doi.org/10.1016/j.jcp.2016.08.042.\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedRK/#OrdinaryDiffEqStabilizedRK.ESERK5","page":"OrdinaryDiffEqStabalizedRK","title":"OrdinaryDiffEqStabilizedRK.ESERK5","text":"ESERK5(; eigen_est = nothing)\n\nStabilized Explicit Method. Fifth order method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.\n\nKeyword Arguments\n\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nJ. Martín-Vaquero, A. Kleefeld. ESERK5: A fifth-order extrapolated stabilized explicit Runge-Kutta method, Journal of Computational and Applied Mathematics, 356, pp 22-36, 2019. doi: https://doi.org/10.1016/j.cam.2019.01.040.\n\n\n\n\n\n","category":"type"},{"location":"explicit/PRK/","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/PRK/#OrdinaryDiffEqPRK","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK","text":"","category":"section"},{"location":"explicit/PRK/","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK","text":"Explicit solvers optimized for a certain number of parallel calls of the system of ordinary differential equations f.","category":"page"},{"location":"explicit/PRK/","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqPRK\", \"KuttaPRK2p5\")","category":"page"},{"location":"explicit/PRK/#Full-list-of-solvers","page":"OrdinaryDiffEqPRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/PRK/","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK","text":"KuttaPRK2p5","category":"page"},{"location":"explicit/PRK/#OrdinaryDiffEqPRK.KuttaPRK2p5","page":"OrdinaryDiffEqPRK","title":"OrdinaryDiffEqPRK.KuttaPRK2p5","text":"KuttaPRK2p5(; thread = OrdinaryDiffEq.True())\n\nExplicit Runge-Kutta Method A 5 parallel, 2 processor method of 5th order.\n\nKeyword Arguments\n\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{jackson1995potential, title={The potential for parallelism in Runge–Kutta methods. Part 1: RK formulas in standard form}, author={Jackson, Kenneth R and Norsett, Syvert Paul}, journal={SIAM journal on numerical analysis}, volume={32}, number={1}, pages={49–82}, year={1995}, publisher={SIAM}}\n\n\n\n\n\n","category":"type"},{"location":"misc/","page":"-","title":"-","text":"SplitEuler\nCompositeAlgorithm","category":"page"},{"location":"explicit/Tsit5/","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/Tsit5/#OrdinaryDiffEqTsit5","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5","text":"","category":"section"},{"location":"explicit/Tsit5/","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5","text":"Recommended solver for most non-stiff problems at default and higher tolerance.","category":"page"},{"location":"explicit/Tsit5/","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqTsit5\", \"Tsit5\")","category":"page"},{"location":"explicit/Tsit5/#Full-list-of-solvers","page":"OrdinaryDiffEqTsit5","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/Tsit5/","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5","text":"Tsit5\nAutoTsit5","category":"page"},{"location":"explicit/Tsit5/#OrdinaryDiffEqTsit5.Tsit5","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5.Tsit5","text":"Tsit5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fifth-order explicit Runge-Kutta method with embedded error estimator of Tsitouras. Free 4th order interpolant.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{tsitouras2011runge, title={Runge–Kutta pairs of order 5 (4) satisfying only the first column simplifying assumption}, author={Tsitouras, Ch}, journal={Computers \\& Mathematics with Applications}, volume={62}, number={2}, pages={770–775}, year={2011}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Tsit5/#OrdinaryDiffEqTsit5.AutoTsit5","page":"OrdinaryDiffEqTsit5","title":"OrdinaryDiffEqTsit5.AutoTsit5","text":"Automatic switching algorithm that can switch between the (non-stiff) Tsit5() and stiff_alg.\n\nAutoTsit5(stiff_alg; kwargs...)\n\nThis method is equivalent to AutoAlgSwitch(Tsit5(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"","category":"section"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"Multistep methods, good for large stiff systems.","category":"page"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqBDF\", \"QNDF\")","category":"page"},{"location":"implicit/BDF/#Full-list-of-solvers","page":"OrdinaryDiffEqBDF","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"ABDF2\nQNDF\nQNDF1\nQNDF2\nQBDF\nQBDF1\nQBDF2\nMEBDF2\nFBDF","category":"page"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.ABDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.ABDF2","text":"ABDF2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n κ = nothing,\n tol = nothing,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :Standard,\n step_limiter! = trivial_limiter!)\n\nMultistep Method. An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ABDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- κ: TBD\ntol: TBD\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nE. Alberdi Celayaa, J. J. Anza Aguirrezabalab, P. Chatzipantelidisc. Implementation of an Adaptive BDF2 Formula and Comparison with The MATLAB Ode15s. Procedia Computer Science, 29, pp 1014-1026, 2014. doi: https://doi.org/10.1016/j.procs.2014.05.091\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QNDF","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QNDF","text":"QNDF(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n κ = nothing,\n tol = nothing,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n kappa = promote(-0.1850, -1 // 9, -0.0823, -0.0415, 0),\n controller = :Standard,\n step_limiter! = trivial_limiter!)\n\nMultistep Method. An adaptive order quasi-constant timestep NDF method. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- κ: TBD\ntol: TBD\nnlsolve: TBD\nextrapolant: TBD\nkappa: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QNDF1","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QNDF1","text":"QNDF1(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n kappa = -0.1850,\n controller = :Standard,\n step_limiter! = trivial_limiter!)\n\nMultistep Method. An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine's accuracy-optimal -0.1850.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF1(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- nlsolve: TBD\nextrapolant: TBD\nkappa: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QNDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QNDF2","text":"QNDF2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n kappa = -1 // 9,\n controller = :Standard,\n step_limiter! = trivial_limiter!)\n\nMultistep Method. An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify QNDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- nlsolve: TBD\nextrapolant: TBD\nkappa: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QBDF","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QBDF","text":"QBDF: Multistep Method\n\nAn alias of QNDF with κ=0.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QBDF1","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QBDF1","text":"QBDF1: Multistep Method\n\nAn alias of QNDF1 with κ=0.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.QBDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.QBDF2","text":"QBDF2: Multistep Method\n\nAn alias of QNDF2 with κ=0.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.MEBDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.MEBDF2","text":"MEBDF2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :constant)\n\nMultistep Method. The second order Modified Extended BDF method, which has improved stability properties over the standard BDF. Fixed timestep only.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify MEBDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- nlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{cash2000modified, title={Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs}, author={Cash, JR}, journal={Journal of Computational and Applied Mathematics}, volume={125}, number={1-2}, pages={117–130}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.FBDF","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.FBDF","text":"FBDF(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n κ = nothing,\n tol = nothing,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :Standard,\n step_limiter! = trivial_limiter!,\n max_order::Val{MO} = Val{5}())\n\nMultistep Method. An adaptive order quasi-constant timestep NDF method. Fixed leading coefficient BDF. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify FBDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- κ: TBD\ntol: TBD\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nmax_order: TBD\n\nReferences\n\n@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH \\& Co. KG}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#IMEX-Multistep","page":"OrdinaryDiffEqBDF","title":"IMEX Multistep","text":"","category":"section"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"SBDF\nSBDF2\nSBDF3\nSBDF4","category":"page"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.SBDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.SBDF2","text":"SBDF2(;kwargs...)\n\nThe two-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nSee also SBDF.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.SBDF3","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.SBDF3","text":"SBDF3(;kwargs...)\n\nThe three-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nSee also SBDF.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.SBDF4","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.SBDF4","text":"SBDF4(;kwargs...)\n\nThe four-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nSee also SBDF.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#IMEX-SDIRK","page":"OrdinaryDiffEqBDF","title":"IMEX SDIRK","text":"","category":"section"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"IMEXEuler\nIMEXEulerARK","category":"page"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.IMEXEuler","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.IMEXEuler","text":"IMEXEuler(;kwargs...)\n\nThe one-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nWhen applied to a SplitODEProblem of the form\n\nu'(t) = f1(u) + f2(u)\n\nThe default IMEXEuler() method uses an update of the form\n\nunew = uold + dt * (f1(unew) + f2(uold))\n\nSee also SBDF, IMEXEulerARK.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.IMEXEulerARK","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.IMEXEulerARK","text":"IMEXEulerARK(;kwargs...)\n\nThe one-step version of the IMEX multistep methods of\n\nUri M. Ascher, Steven J. Ruuth, Brian T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. Society for Industrial and Applied Mathematics. Journal on Numerical Analysis, 32(3), pp 797-823, 1995. doi: https://doi.org/10.1137/0732037\n\nWhen applied to a SplitODEProblem of the form\n\nu'(t) = f1(u) + f2(u)\n\nA classical additive Runge-Kutta method in the sense of Araújo, Murua, Sanz-Serna (1997) consisting of the implicit and the explicit Euler method given by\n\ny1 = uold + dt * f1(y1)\nunew = uold + dt * (f1(unew) + f2(y1))\n\nSee also SBDF, IMEXEuler.\n\n\n\n\n\n","category":"function"},{"location":"implicit/BDF/#DAE","page":"OrdinaryDiffEqBDF","title":"DAE","text":"","category":"section"},{"location":"implicit/BDF/","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF","text":"DImplicitEuler\nDABDF2\nDFBDF","category":"page"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.DImplicitEuler","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.DImplicitEuler","text":"DImplicitEuler(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :constant,\n controller = :Standard)\n\nMultistep Method. Implicit Euler for implicit DAE form. It uses an apriori error estimator for adaptivity based on a finite differencing approximation from SPICE.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DImplicitEuler(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- nlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.DABDF2","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.DABDF2","text":"DABDF2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :constant,\n controller = :Standard)\n\nMultistep Method. Fully implicit implementation of BDF2.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DABDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- nlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{celaya2014implementation, title={Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s}, author={Celaya, E Alberdi and Aguirrezabala, JJ Anza and Chatzipantelidis, Panagiotis}, journal={Procedia Computer Science}, volume={29}, pages={1014–1026}, year={2014}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/BDF/#OrdinaryDiffEqBDF.DFBDF","page":"OrdinaryDiffEqBDF","title":"OrdinaryDiffEqBDF.DFBDF","text":"DFBDF(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n κ = nothing,\n tol = nothing,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :Standard,\n max_order::Val{MO} = Val{5}())\n\nMultistep Method. Fully implicit implementation of FBDF based on Shampine's\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify DFBDF(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n/n- κ: TBD\ntol: TBD\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\nmax_order: TBD\n\nReferences\n\n@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH and Co. KG} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/HighOrderRK/","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/HighOrderRK/#OrdinaryDiffEqHighOrderRK","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK","text":"","category":"section"},{"location":"explicit/HighOrderRK/","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK","text":"Solvers for non-stiff problems at low tolerance. However, the solvers in OrdinaryDiffEqVerner generally perform better at low tolerances.","category":"page"},{"location":"explicit/HighOrderRK/","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqHighOrderRK\", \"DP8\")","category":"page"},{"location":"explicit/HighOrderRK/#Full-list-of-solvers","page":"OrdinaryDiffEqHighOrderRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/HighOrderRK/","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK","text":"TanYam7\nTsitPap8\nDP8\nPFRK87","category":"page"},{"location":"explicit/HighOrderRK/#OrdinaryDiffEqHighOrderRK.TanYam7","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK.TanYam7","text":"TanYam7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Tanaka-Yamashita 7 Runge-Kutta method. (7th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nTanaka M., Muramatsu S., Yamashita S., (1992), On the Optimization of Some Nine-Stage Seventh-order Runge-Kutta Method, Information Processing Society of Japan, 33 (12), pp. 1512-1526.\n\n\n\n\n\n","category":"type"},{"location":"explicit/HighOrderRK/#OrdinaryDiffEqHighOrderRK.TsitPap8","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK.TsitPap8","text":"TsitPap8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Tsitouras-Papakostas 8/7 Runge-Kutta method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{tsitouras1999cheap, title={Cheap error estimation for Runge–Kutta methods}, author={Tsitouras, Ch and Papakostas, SN}, journal={SIAM Journal on Scientific Computing}, volume={20}, number={6}, pages={2067–2088}, year={1999}, publisher={SIAM}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/HighOrderRK/#OrdinaryDiffEqHighOrderRK.DP8","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK.DP8","text":"DP8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Hairer's 8/5/3 adaption of the Dormand-Prince Runge-Kutta method. (7th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/HighOrderRK/#OrdinaryDiffEqHighOrderRK.PFRK87","page":"OrdinaryDiffEqHighOrderRK","title":"OrdinaryDiffEqHighOrderRK.PFRK87","text":"PFRK87(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n omega = 0.0)\n\nExplicit Runge-Kutta Method. Phase-fitted Runge-Kutta of 8th order.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nomega: a periodicity phase estimate, when accurate this method results in zero numerical dissipation.\n\nReferences\n\n@article{tsitouras2017phase, title={Phase-fitted Runge–Kutta pairs of orders 8 (7)}, author={Tsitouras, Ch and Famelis, I Th and Simos, TE}, journal={Journal of Computational and Applied Mathematics}, volume={321}, pages={226–231}, year={2017}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"","category":"section"},{"location":"implicit/SDIRK/","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"This article is a stub.","category":"page"},{"location":"implicit/SDIRK/","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqSDIRK\", \"PDIRK44\")","category":"page"},{"location":"implicit/SDIRK/#Full-list-of-solvers","page":"OrdinaryDiffEqSDIRK","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/SDIRK/","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"ImplicitEuler\nImplicitMidpoint\nTrapezoid\nTRBDF2\nSDIRK2\nSDIRK22\nSSPSDIRK2\nKvaerno3\nKenCarp3\nCFNLIRK3\nCash4\nSFSDIRK4\nSFSDIRK5\nSFSDIRK6\nSFSDIRK7\nSFSDIRK8\nHairer4\nHairer42\nKvaerno4\nKvaerno5","category":"page"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ImplicitEuler","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ImplicitEuler","text":"ImplicitEuler(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :constant,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. A 1st order implicit solver. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitEuler(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@book{wanner1996solving, title={Solving ordinary differential equations II}, author={Wanner, Gerhard and Hairer, Ernst}, volume={375}, year={1996}, publisher={Springer Berlin Heidelberg New York}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ImplicitMidpoint","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ImplicitMidpoint","text":"ImplicitMidpoint(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ImplicitMidpoint(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@book{wanner1996solving, title={Solving ordinary differential equations II}, author={Wanner, Gerhard and Hairer, Ernst}, volume={375}, year={1996}, publisher={Springer Berlin Heidelberg New York}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Trapezoid","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Trapezoid","text":"Trapezoid(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. Second order A-stable symmetric ESDIRK method. \"Almost symplectic\" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences approximation to the second derivative terms in the local truncation error estimate (the SPICE approximation strategy).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Trapezoid(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nAndre Vladimirescu. 1994. The Spice Book. John Wiley & Sons, Inc., New York, NY, USA.\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.TRBDF2","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.TRBDF2","text":"TRBDF2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. A second order A-B-L-S-stable one-step ESDIRK method. Includes stiffness-robust error estimates for accurate adaptive timestepping, smoothed derivatives for highly stiff and oscillatory problems.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify TRBDF2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{hosea1996analysis, title={Analysis and implementation of TR-BDF2}, author={Hosea, ME and Shampine, LF}, journal={Applied Numerical Mathematics}, volume={20}, number={1-2}, pages={21–37}, year={1996}, publisher={Elsevier}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SDIRK2","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SDIRK2","text":"SDIRK2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. SDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SDIRK2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SDIRK22","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SDIRK22","text":"SDIRK22(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. Description TBD\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SDIRK22(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@techreport{kennedy2016diagonally, title={Diagonally implicit Runge-Kutta methods for ordinary differential equations. A review}, author={Kennedy, Christopher A and Carpenter, Mark H}, year={2016}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SSPSDIRK2","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SSPSDIRK2","text":"SSPSDIRK2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :constant,\n controller = :PI)\n\nSDIRK Method. SSPSDIRK is an SSP-optimized SDIRK method, so it's an implicit SDIRK method for handling stiffness but if the dt is below the SSP coefficient * dt, then the SSP property of the SSP integrators (the other page) is satisified. As such this is a method which is expected to be good on advection-dominated cases where an explicit SSP integrator would be used, but where reaction equations are sufficient stiff to justify implicit integration.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SSPSDIRK2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{ketcheson2009optimal, title={Optimal implicit strong stability preserving Runge–Kutta methods}, author={Ketcheson, David I and Macdonald, Colin B and Gottlieb, Sigal}, journal={Applied Numerical Mathematics}, volume={59}, number={2}, pages={373–392}, year={2009}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Kvaerno3","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Kvaerno3","text":"Kvaerno3(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 3rd order ESDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.KenCarp3","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.KenCarp3","text":"KenCarp3(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.CFNLIRK3","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.CFNLIRK3","text":"CFNLIRK3(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Third order method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify CFNLIRK3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{calvo2001linearly, title={Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations}, author={Calvo, MP and De Frutos, J and Novo, J}, journal={Applied Numerical Mathematics}, volume={37}, number={4}, pages={535–549}, year={2001}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Cash4","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Cash4","text":"Cash4(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n embedding = 3)\n\nSDIRK Method. An A-L stable 4th order SDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Cash4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nembedding: TBD\n\nReferences\n\n@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SFSDIRK4","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SFSDIRK4","text":"SFSDIRK4(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Method of order 4.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SFSDIRK5","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SFSDIRK5","text":"SFSDIRK5(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Method of order 5.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SFSDIRK6","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SFSDIRK6","text":"SFSDIRK6(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Method of order 6.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK6(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SFSDIRK7","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SFSDIRK7","text":"SFSDIRK7(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Method of order 7.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK7(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.SFSDIRK8","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.SFSDIRK8","text":"SFSDIRK8(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear)\n\nSDIRK Method. Method of order 8.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify SFSDIRK8(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\n\nReferences\n\n@article{ferracina2008strong, title={Strong stability of singly-diagonally-implicit Runge–Kutta methods}, author={Ferracina, Luca and Spijker, MN}, journal={Applied Numerical Mathematics}, volume={58}, number={11}, pages={1675–1686}, year={2008}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Hairer4","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Hairer4","text":"Hairer4(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. An A-L stable 4th order SDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Hairer4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Hairer42","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Hairer42","text":"Hairer42(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. An A-L stable 4th order SDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Hairer42(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Kvaerno4","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Kvaerno4","text":"Kvaerno4(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 4th order ESDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter: TBD\n\nReferences\n\n@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.Kvaerno5","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.Kvaerno5","text":"Kvaerno5(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 5th order ESDIRK method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Kvaerno5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter: TBD\n\nReferences\n\n@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\\ae}rn{\\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#IMEX-SDIRK","page":"OrdinaryDiffEqSDIRK","title":"IMEX SDIRK","text":"","category":"section"},{"location":"implicit/SDIRK/","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK","text":"KenCarp4\nKenCarp47\nKenCarp5\nKenCarp58\nESDIRK54I8L2SA\nESDIRK436L2SA2\nESDIRK437L2SA\nESDIRK547L2SA2\nESDIRK659L2SA","category":"page"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.KenCarp4","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.KenCarp4","text":"KenCarp4(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 4th order ESDIRK method with splitting.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp4(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter: TBD\n\nReferences\n\n@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.KenCarp47","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.KenCarp47","text":"KenCarp47(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp47(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.KenCarp5","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.KenCarp5","text":"KenCarp5(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI,\n step_limiter! = trivial_limiter!)\n\nSDIRK Method. An A-L stable stiffly-accurate 5th order ESDIRK method with splitting.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\nstep_limiter: TBD\n\nReferences\n\n@book{kennedy2001additive, title={Additive Runge-Kutta schemes for convection-diffusion-reaction equations}, author={Kennedy, Christopher Alan}, year={2001}, publisher={National Aeronautics and Space Administration, Langley Research Center}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.KenCarp58","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.KenCarp58","text":"KenCarp58(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n smooth_est = true,\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify KenCarp58(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nsmooth_est: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{kennedy2019higher, title={Higher-order additive Runge–Kutta schemes for ordinary differential equations}, author={Kennedy, Christopher A and Carpenter, Mark H}, journal={Applied Numerical Mathematics}, volume={136}, pages={183–205}, year={2019}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ESDIRK54I8L2SA","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ESDIRK54I8L2SA","text":"ESDIRK54I8L2SA(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK54I8L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ESDIRK436L2SA2","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ESDIRK436L2SA2","text":"ESDIRK436L2SA2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK436L2SA2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ESDIRK437L2SA","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ESDIRK437L2SA","text":"ESDIRK437L2SA(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK437L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ESDIRK547L2SA2","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ESDIRK547L2SA2","text":"ESDIRK547L2SA2(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK547L2SA2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"implicit/SDIRK/#OrdinaryDiffEqSDIRK.ESDIRK659L2SA","page":"OrdinaryDiffEqSDIRK","title":"OrdinaryDiffEqSDIRK.ESDIRK659L2SA","text":"ESDIRK659L2SA(; chunk_size = Val{0}(),\n autodiff = true,\n standardtag = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:forward},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n nlsolve = NLNewton(),\n extrapolant = :linear,\n controller = :PI)\n\nSDIRK Method. Optimized ESDIRK tableaus. Updates of the original KenCarp tableau expected to achieve lower error for the same steps in theory, but are still being fully evaluated in context. Currently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ESDIRK659L2SA(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nnlsolve: TBD\nextrapolant: TBD\ncontroller: TBD\n\nReferences\n\n@article{Kennedy2019DiagonallyIR, title={Diagonally implicit Runge–Kutta methods for stiff ODEs}, author={Christopher A. Kennedy and Mark H. Carpenter}, journal={Applied Numerical Mathematics}, year={2019}, volume={146}, pages={221-244} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Nordsieck/","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/Nordsieck/#OrdinaryDiffEqNordsieck","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"","category":"section"},{"location":"explicit/Nordsieck/","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"The Nordsieck form is an alternative representation of multistep methods which, instead of representing and saving past step values in a history vector, it uses a derivative list (like a Taylor expansion) for the computation of the next point. The Nordsieck form was pioneered by early implementations of BDF methods such LSODE, VODE, and finally CVODE. It can have some advantages in terms of restartability as the full Nordsieck vector can be instantiated given only the information of f and its derivatives after discontinuities, but the higher derivative representations can also introduce numerical instabilities of their own.","category":"page"},{"location":"explicit/Nordsieck/","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"The Nordsieck implementations here are considered experimental implementations of the LSODE non-fixed leading coefficient form and are generally considered inferior to the fixed-leading history-based BDF implementation of FBDF, and thus for all standard usage we recommend FBDF. However, this algorithm is kept for experimental research and development purposes with the possibility of one day becoming a more discontinuity-aware BDF implementation.","category":"page"},{"location":"explicit/Nordsieck/","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqNordsieck\", \"AN5\")","category":"page"},{"location":"explicit/Nordsieck/#Full-list-of-solvers","page":"OrdinaryDiffEqNordsieck","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/Nordsieck/","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck","text":"AN5\nJVODE\nJVODE_Adams\nJVODE_BDF","category":"page"},{"location":"explicit/Nordsieck/#OrdinaryDiffEqNordsieck.AN5","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck.AN5","text":"AN5()\n\nAdaptive step size Adams explicit Method An adaptive 5th order fixed-leading coefficient Adams method in Nordsieck form.\n\nwarning: Experimental\nAN5 is experimental, the solver VCABM is generally preferred.\n\nKeyword Arguments\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"explicit/Nordsieck/#OrdinaryDiffEqNordsieck.JVODE","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck.JVODE","text":"warning: Experimental\nJVODE is experimental, the solver VCABM is generally preferred.\n\n\n\n\n\n","category":"type"},{"location":"explicit/Nordsieck/#OrdinaryDiffEqNordsieck.JVODE_Adams","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck.JVODE_Adams","text":"warning: Experimental\nJVODE is experimental, the solver VCABM is generally preferred.\n\n\n\n\n\n","category":"function"},{"location":"explicit/Nordsieck/#OrdinaryDiffEqNordsieck.JVODE_BDF","page":"OrdinaryDiffEqNordsieck","title":"OrdinaryDiffEqNordsieck.JVODE_BDF","text":"warning: Experimental\nJVODE is experimental, the solver FBDF is generally preferred.\n\n\n\n\n\n","category":"function"},{"location":"implicit/Rosenbrock/","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock","text":"","category":"section"},{"location":"implicit/Rosenbrock/","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock","text":"Methods for small and medium sized stiff systems of differential equations. At high tolerances, >1e-2, Rosenbrock23 is a good choice. At medium tolerances >1e-8 it is recommended you use Rodas5P or Rodas4P, the former is more efficient, but the latter is more reliable. For larger systems look at multistep methods.","category":"page"},{"location":"implicit/Rosenbrock/","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqRosenbrock\", \"Rodas5P\")","category":"page"},{"location":"implicit/Rosenbrock/#Full-list-of-solvers","page":"OrdinaryDiffEqRosenbrock","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/Rosenbrock/","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock","text":"Rosenbrock23\nRosenbrock32\nROS3P\nRodas3\nRodas23W\nRodas3P\nRodas4\nRodas42\nRodas4P\nRodas4P2\nRodas5\nRodas5P\nRodas5Pe\nRodas5Pr\nRosenbrockW6S4OS\nROS2\nROS2PR\nROS2S\nROS3\nROS3PR\nScholz4_7\nROS34PW1a\nROS34PW1b\nROS34PW2\nROS34PW3\nROS34PRw\nROS3PRL\nROS3PRL2\nROK4a\nRosShamp4\nVeldd4\nVelds4\nGRK4T\nGRK4A\nRos4LStab","category":"page"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rosenbrock23","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rosenbrock23","text":"Rosenbrock23(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock-Wanner Method. An Order 2/3 L-Stable Rosenbrock-W method which is good for very stiff equations with oscillations at low tolerances. 2nd order stiff-aware interpolation.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rosenbrock23(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nShampine L.F. and Reichelt M., (1997) The MATLAB ODE Suite, SIAM Journal of\n\nScientific Computing, 18 (1), pp. 1-22.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rosenbrock32","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rosenbrock32","text":"Rosenbrock32(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock-Wanner Method. An Order 3/2 A-Stable Rosenbrock-W method which is good for mildly stiff equations without oscillations at low tolerances. Note that this method is prone to instability in the presence of oscillations, so use with caution. 2nd order stiff-aware interpolation.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rosenbrock32(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nShampine L.F. and Reichelt M., (1997) The MATLAB ODE Suite, SIAM Journal of\n\nScientific Computing, 18 (1), pp. 1-22.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS3P","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS3P","text":"ROS3P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS3P(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method. Keeps high accuracy on discretizations of nonlinear parabolic PDEs.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nLang, J. & Verwer, ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems J. BIT Numerical Mathematics (2001) 41: 731. doi:10.1023/A:1021900219772\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas3","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas3","text":"Rodas3(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas3(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G. Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package. In: BIT Numerical Mathematics, 63(2), 2023\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas23W","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas23W","text":"Rodas23W(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock-Wanner Method. An Order 2/3 L-Stable Rosenbrock-W method for stiff ODEs and DAEs in mass matrix form. 2nd order stiff-aware interpolation and additional error test for interpolation.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas23W(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas3P","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas3P","text":"Rodas3P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas3P(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. 3rd order A-stable and stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant and additional error test for interpolation. Keeps accuracy on discretizations of linear parabolic PDEs.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas4","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas4","text":"Rodas4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas4(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 4th order L-stable Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas42","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas42","text":"Rodas42(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas42(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas4P","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas4P","text":"Rodas4P(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock-Wanner Method. 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems (as opposed to lower if not corrected).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas4P(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas4P2","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas4P2","text":"Rodas4P2(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock-Wanner Method. A 4th order L-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems. It is an improvement of Roadas4P and in case of inexact Jacobians a second order W method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify Rodas4P2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G., Rodas23W / Rodas32P - a Rosenbrock-type method for DAEs with additional error estimate for dense output and Julia implementation, In progress.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas5","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas5","text":"Rodas5(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas5(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nDi Marzo G. RODAS5(4) – Méthodes de Rosenbrock d’ordre 5(4) adaptées aux problemes différentiels-algébriques. MSc mathematics thesis, Faculty of Science, University of Geneva, Switzerland.\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas5P","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas5P","text":"Rodas5P(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas5P(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G. Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package. In: BIT Numerical Mathematics, 63(2), 2023\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas5Pe","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas5Pe","text":"Rodas5Pe(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas5Pe(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G. Rosenbrock methods within OrdinaryDiffEq.jl - Overview, recent developments and applications - Preprint 2024 https://github.com/hbrs-cse/RosenbrockMethods/blob/main/paper/JuliaPaper.pdf\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Rodas5Pr","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Rodas5Pr","text":"Rodas5Pr(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Rodas5Pr(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nSteinebach G. Rosenbrock methods within OrdinaryDiffEq.jl - Overview, recent developments and applications - Preprint 2024 https://github.com/hbrs-cse/RosenbrockMethods/blob/main/paper/JuliaPaper.pdf\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.RosenbrockW6S4OS","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.RosenbrockW6S4OS","text":"RosenbrockW6S4OS(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method (fixed step only).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RosenbrockW6S4OS(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\nhttps://doi.org/10.1016/j.cam.2009.09.017\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS2","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS2","text":"ROS2(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS2(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. A 2nd order L-stable Rosenbrock method with 2 internal stages.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nJ. G. Verwer et al. (1999): A second-order Rosenbrock method applied to photochemical dispersion problems https://doi.org/10.1137/S1064827597326651\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS2PR","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS2PR","text":"ROS2PR(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS2PR(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 2nd order stiffly accurate Rosenbrock method with 3 internal stages with (Rinf=0). For problems with medium stiffness the convergence behaviour is very poor and it is recommended to use ROS2S instead.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS2S","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS2S","text":"ROS2S(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. 2nd order stiffly accurate Rosenbrock-Wanner W-method with 3 internal stages with B_PR consistent of order 2 with (Rinf=0).\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS2S(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS3","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS3","text":"ROS3(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS3(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 3rd order L-stable Rosenbrock method with 3 internal stages with an embedded strongly A-stable 2nd order method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS3PR","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS3PR","text":"ROS3PR(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS3PR(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 3nd order stiffly accurate Rosenbrock method with 3 internal stages with B_PR consistent of order 3, which is strongly A-stable with Rinf~=-0.73.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Scholz4_7","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Scholz4_7","text":"Scholz4_7(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Scholz4_7(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 3nd order stiffly accurate Rosenbrock method with 3 internal stages with B_PR consistent of order 3, which is strongly A-stable with Rinf~=-0.73. Convergence with order 4 for the stiff case, but has a poor accuracy.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS34PW1a","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS34PW1a","text":"ROS34PW1a(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW1a(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\n@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS34PW1b","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS34PW1b","text":"ROS34PW1b(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. A 4th order L-stable Rosenbrock-W method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW1b(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\n@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS34PW2","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS34PW2","text":"ROS34PW2(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. A 4th order stiffy accurate Rosenbrock-W method for PDAEs.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW2(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\n@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS34PW3","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS34PW3","text":"ROS34PW3(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. A 4th order strongly A-stable (Rinf~0.63) Rosenbrock-W method.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PW3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\n@article{rang2005new, title={New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1}, author={Rang, Joachim and Angermann, L}, journal={BIT Numerical Mathematics}, volume={45}, pages={761–787}, year={2005}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS34PRw","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS34PRw","text":"ROS34PRw(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. 3rd order stiffly accurate Rosenbrock-Wanner W-method with 4 internal stages, B_PR consistent of order 2. The order of convergence decreases if medium stiff problems are considered.\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROS34PRw(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\nJoachim Rang, Improved traditional Rosenbrock–Wanner methods for stiff ODEs and DAEs, Journal of Computational and Applied Mathematics, https://doi.org/10.1016/j.cam.2015.03.010\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS3PRL","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS3PRL","text":"ROS3PRL(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS3PRL(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 3rd order stiffly accurate Rosenbrock method with 4 internal stages, B_PR consistent of order 2 with Rinf=0. The order of convergence decreases if medium stiff problems are considered, but it has good results for very stiff cases.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROS3PRL2","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROS3PRL2","text":"ROS3PRL2(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `ROS3PRL2(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. 3rd order stiffly accurate Rosenbrock method with 4 internal stages, B_PR consistent of order 3. The order of convergence does NOT decreases if medium stiff problems are considered as it does for ROS3PRL.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nRang, Joachim (2014): The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods. https://doi.org/10.24355/dbbs.084-201408121139-0\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.ROK4a","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.ROK4a","text":"ROK4a(; chunk_size = Val{0}(),\n standardtag = Val{true}(),\n autodiff = Val{true}(),\n concrete_jac = nothing,\n diff_type = Val{:central},\n linsolve = nothing,\n precs = DEFAULT_PRECS)\n\nRosenbrock-Wanner Method. 4rd order L-stable Rosenbrock-Krylov method with 4 internal stages, with a 3rd order embedded method which is strongly A-stable with Rinf~=0.55. (when using exact Jacobians)\n\nKeyword Arguments\n\nchunk_size: The chunk size used with ForwardDiff.jl. Defaults to Val{0}() and thus uses the internal ForwardDiff.jl algorithm for the choice.\nstandardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().\nautodiff: Specifies whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to Val{true}() for automatic differentiation.\nconcrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.\ndiff_type: The type of differentiation used in FiniteDiff.jl if autodiff=false. Defaults to Val{:forward}, with alternatives of Val{:central} and Val{:complex}.\nlinsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify ROK4a(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.\nprecs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:\nW: the current Jacobian of the nonlinear system. Specified as either I - gamma J or Igamma - J depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.\ndu: the current ODE derivative\nu: the current ODE state\np: the ODE parameters\nt: the current ODE time\nnewW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.\nPlprev: the previous Pl.\nPrprev: the previous Pr.\nsolverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.\nThe return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:\nPl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\nwhich is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:\nDEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n\nReferences\n\nTranquilli, Paul and Sandu, Adrian (2014): Rosenbrock–Krylov Methods for Large Systems of Differential Equations https://doi.org/10.1137/130923336\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.RosShamp4","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.RosShamp4","text":"RosShamp4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `RosShamp4(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```)\n\nRosenbrock Method. An A-stable 4th order Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\n\nReferences\n\nL. F. Shampine, Implementation of Rosenbrock Methods, ACM Transactions on Mathematical Software (TOMS), 8: 2, 93-113. doi:10.1145/355993.355994\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Veldd4","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Veldd4","text":"Veldd4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Veldd4(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 4th order D-stable Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nvan Veldhuizen, D-stability and Kaps-Rentrop-methods, M. Computing (1984) 32: 229. doi:10.1007/BF02243574\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Velds4","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Velds4","text":"Velds4(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Velds4(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 4th order A-stable Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nvan Veldhuizen, D-stability and Kaps-Rentrop-methods, M. Computing (1984) 32: 229. doi:10.1007/BF02243574\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.GRK4T","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.GRK4T","text":"GRK4T(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `GRK4T(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. An efficient 4th order Rosenbrock method.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nKaps, P. & Rentrop, Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. P. Numer. Math. (1979) 33: 55. doi:10.1007/BF01396495\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.GRK4A","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.GRK4A","text":"GRK4A(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `GRK4A(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. An A-stable 4th order Rosenbrock method. Essentially \"anti-L-stable\" but efficient.\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nKaps, P. & Rentrop, Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. P. Numer. Math. (1979) 33: 55. doi:10.1007/BF01396495\n\n\n\n\n\n","category":"type"},{"location":"implicit/Rosenbrock/#OrdinaryDiffEqRosenbrock.Ros4LStab","page":"OrdinaryDiffEqRosenbrock","title":"OrdinaryDiffEqRosenbrock.Ros4LStab","text":"Ros4LStab(; - `chunk_size`: The chunk size used with ForwardDiff.jl. Defaults to `Val{0}()`\n and thus uses the internal ForwardDiff.jl algorithm for the choice.\n - `standardtag`: Specifies whether to use package-specific tags instead of the\n ForwardDiff default function-specific tags. For more information, see\n [this blog post](https://www.stochasticlifestyle.com/improved-forwarddiff-jl-stacktraces-with-package-tags/).\n Defaults to `Val{true}()`.\n - `autodiff`: Specifies whether to use automatic differentiation via\n [ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) or finite\n differencing via [FiniteDiff.jl](https://github.com/JuliaDiff/FiniteDiff.jl).\n Defaults to `Val{true}()` for automatic differentiation.\n - `concrete_jac`: Specifies whether a Jacobian should be constructed. Defaults to\n `nothing`, which means it will be chosen true/false depending on circumstances\n of the solver, such as whether a Krylov subspace method is used for `linsolve`.\n - `diff_type`: The type of differentiation used in FiniteDiff.jl if `autodiff=false`.\n Defaults to `Val{:forward}`, with alternatives of `Val{:central}` and\n `Val{:complex}`.\n - `linsolve`: Any [LinearSolve.jl](https://github.com/SciML/LinearSolve.jl) compatible linear solver.\n For example, to use [KLU.jl](https://github.com/JuliaSparse/KLU.jl), specify\n `Ros4LStab(linsolve = KLUFactorization()`).\n When `nothing` is passed, uses `DefaultLinearSolver`.\n - `precs`: Any [LinearSolve.jl-compatible preconditioner](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/)\n can be used as a left or right preconditioner.\n Preconditioners are specified by the `Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata)`\n function where the arguments are defined as:\n - `W`: the current Jacobian of the nonlinear system. Specified as either\n ``I - \\gamma J`` or ``I/\\gamma - J`` depending on the algorithm. This will\n commonly be a `WOperator` type defined by OrdinaryDiffEq.jl. It is a lazy\n representation of the operator. Users can construct the W-matrix on demand\n by calling `convert(AbstractMatrix,W)` to receive an `AbstractMatrix` matching\n the `jac_prototype`.\n - `du`: the current ODE derivative\n - `u`: the current ODE state\n - `p`: the ODE parameters\n - `t`: the current ODE time\n - `newW`: a `Bool` which specifies whether the `W` matrix has been updated since\n the last call to `precs`. It is recommended that this is checked to only\n update the preconditioner when `newW == true`.\n - `Plprev`: the previous `Pl`.\n - `Prprev`: the previous `Pr`.\n - `solverdata`: Optional extra data the solvers can give to the `precs` function.\n Solver-dependent and subject to change.\n The return is a tuple `(Pl,Pr)` of the LinearSolve.jl-compatible preconditioners.\n To specify one-sided preconditioning, simply return `nothing` for the preconditioner\n which is not used. Additionally, `precs` must supply the dispatch:\n ```julia\n Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)\n ```\n which is used in the solver setup phase to construct the integrator\n type with the preconditioners `(Pl,Pr)`.\n The default is `precs=DEFAULT_PRECS` where the default preconditioner function\n is defined as:\n ```julia\n DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing\n ```\n step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nRosenbrock Method. A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant\n\nKeyword Arguments\n\nchunk_size: TBD\nstandardtag: TBD\nautodiff: boolean to control if the Jacobian should be computed via AD or not\nconcrete_jac: function of the form jac!(J, u, p, t)\ndiff_type: TBD\nlinsolve: custom solver for the inner linear systems\nprecs: custom preconditioner for the inner linear solver\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\nE. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK","text":"","category":"section"},{"location":"explicit/LowOrderRK/","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK","text":"If OrdinaryDiffEqTsit5 is not working well for your non-stiff problem at default and higher tolerance, it can be worthwhile to explore the options in this package. In particular, when more robust error control is required, BS5 is a good choice. If at moderate tolerances and the interpolation error is very important, consider the OwrenZen5 method. For fast solving at higher tolerances, we recommend BS3, or OwrenZen3if the interpolation error is important.","category":"page"},{"location":"explicit/LowOrderRK/","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqLowOrderRK\", \"BS3\")","category":"page"},{"location":"explicit/LowOrderRK/#Full-list-of-solvers","page":"OrdinaryDiffEqLowOrderRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/LowOrderRK/","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK","text":"Euler\nHeun\nRalston\nMidpoint\nRK4\nBS3\nOwrenZen3\nOwrenZen4\nOwrenZen5\nBS5\nDP5\nAnas5\nRKO65\nFRK65\nRKM\nMSRK5\nMSRK6\nPSRK4p7q6\nPSRK3p5q4\nPSRK3p6q5\nStepanov5\nSIR54\nAlshina2\nAlshina3\nAlshina6","category":"page"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Euler","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Euler","text":"Euler()\n\nExplicit Runge-Kutta Method. The canonical forward Euler method. Fixed timestep only.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Heun","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Heun","text":"Heun(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. The second order Heun's method. Uses embedded Euler method for adaptivity.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Ralston","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Ralston","text":"Ralston(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. The optimized second order midpoint method. Uses embedded Euler method for adaptivity.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Midpoint","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Midpoint","text":"Midpoint(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. The second order midpoint method. Uses embedded Euler method for adaptivity.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.RK4","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.RK4","text":"RK4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. The canonical Runge-Kutta Order 4 method. Uses a defect control for adaptive stepping using maximum error over the whole interval.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{shampine2005solving, title={Solving ODEs and DDEs with residual control}, author={Shampine, LF}, journal={Applied Numerical Mathematics}, volume={52}, number={1}, pages={113–127}, year={2005}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.BS3","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.BS3","text":"BS3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, four-stage FSAL method with embedded error estimator of Bogacki and Shampine.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{bogacki19893, title={A 3 (2) pair of Runge-Kutta formulas}, author={Bogacki, Przemyslaw and Shampine, Lawrence F}, journal={Applied Mathematics Letters}, volume={2}, number={4}, pages={321–325}, year={1989}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.OwrenZen3","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.OwrenZen3","text":"OwrenZen3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 3/2 method (free 3rd order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.OwrenZen4","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.OwrenZen4","text":"OwrenZen4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 4/3 method (free 4th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.OwrenZen5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.OwrenZen5","text":"OwrenZen5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Owren-Zennaro optimized interpolation 5/4 method (free 5th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{owren1992derivation, title={Derivation of efficient, continuous, explicit Runge–Kutta methods}, author={Owren, Brynjulf and Zennaro, Marino}, journal={SIAM journal on scientific and statistical computing}, volume={13}, number={6}, pages={1488–1501}, year={1992}, publisher={SIAM} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.BS5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.BS5","text":"BS5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n lazy = true)\n\nExplicit Runge-Kutta Method. Bogacki-Shampine 5/4 Runge-Kutta method. (lazy 5th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nlazy: determines if the lazy interpolant is used.\n\nReferences\n\n@article{bogacki1996efficient, title={An efficient runge-kutta (4, 5) pair}, author={Bogacki, P and Shampine, Lawrence F}, journal={Computers \\& Mathematics with Applications}, volume={32}, number={6}, pages={15–28}, year={1996}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.DP5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.DP5","text":"DP5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Dormand-Prince's 5/4 Runge-Kutta method. (free 4th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{dormand1980family, title={A family of embedded Runge-Kutta formulae}, author={Dormand, John R and Prince, Peter J}, journal={Journal of computational and applied mathematics}, volume={6}, number={1}, pages={19–26}, year={1980}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Anas5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Anas5","text":"Anas5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n w = 1)\n\nExplicit Runge-Kutta Method. 4th order Runge-Kutta method designed for periodic problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nw: a periodicity estimate, which when accurate the method becomes 5th order\n\n(and is otherwise 4th order with less error for better estimates).\n\nReferences\n\n@article{anastassi2005optimized, title={An optimized Runge–Kutta method for the solution of orbital problems}, author={Anastassi, ZA and Simos, TE}, journal={Journal of Computational and Applied Mathematics}, volume={175}, number={1}, pages={1–9}, year={2005}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.RKO65","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.RKO65","text":"RKO65(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 5th order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nTsitouras, Ch. \"Explicit Runge–Kutta methods for starting integration of Lane–Emden problem.\" Applied Mathematics and Computation 354 (2019): 353-364. doi: https://doi.org/10.1016/j.amc.2019.02.047\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.FRK65","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.FRK65","text":"FRK65(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n omega = 0.0)\n\nExplicit Runge-Kutta Method. Zero Dissipation Runge-Kutta of 6th order.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nomega: a periodicity phase estimate,\n\nwhen accurate this method results in zero numerical dissipation.\n\nReferences\n\n@article{medvedev2018fitted, title={Fitted modifications of Runge-Kutta pairs of orders 6 (5)}, author={Medvedev, Maxim A and Simos, TE and Tsitouras, Ch}, journal={Mathematical Methods in the Applied Sciences}, volume={41}, number={16}, pages={6184–6194}, year={2018}, publisher={Wiley Online Library}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.RKM","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.RKM","text":"RKM(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Method designed to have good stability properties when applied to pseudospectral discretizations of hyperbolic partial differential equaitons.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{mead1999optimal, title={Optimal Runge–Kutta methods for first order pseudospectral operators}, author={Mead, JL and Renaut, RA}, journal={Journal of Computational Physics}, volume={152}, number={1}, pages={404–419}, year={1999}, publisher={Elsevier} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.MSRK5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.MSRK5","text":"MSRK5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 5th order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nMisha Stepanov - https://arxiv.org/pdf/2202.08443.pdf : Figure 3.\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.MSRK6","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.MSRK6","text":"MSRK6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 6th order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nMisha Stepanov - https://arxiv.org/pdf/2202.08443.pdf : Table4\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.PSRK4p7q6","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.PSRK4p7q6","text":"PSRK4p7q6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 6-stage Pseudo-Symplectic method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, volume = {38}, PAGES = {439-461}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, volume = {328}, pages = {86-94}, year = {2017}, issn = {0021-9991}, doi = {https://doi.org/10.1016/j.jcp.2016.10.040}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.PSRK3p5q4","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.PSRK3p5q4","text":"PSRK3p5q4(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 4-stage Pseudo-Symplectic method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, year = {2017}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.PSRK3p6q5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.PSRK3p6q5","text":"PSRK3p6q5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 5-stage Pseudo-Symplectic method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Aubry1998, author = {A. Aubry and P. Chartier}, journal = {BIT Numer. Math.}, title = {Pseudo-symplectic {R}unge-{K}utta methods}, year = {1998}, }, @article{Capuano2017, title = {Explicit {R}unge–{K}utta schemes for incompressible flow with improved energy-conservation properties}, journal = {J. Comput. Phys.}, year = {2017}, author = {F. Capuano and G. Coppola and L. Rández and L. {de Luca}},}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Stepanov5","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Stepanov5","text":"Stepanov5(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 5th order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Stepanov2021Embedded5, title={Embedded (4, 5) pairs of explicit 7-stage Runge–Kutta methods with FSAL property}, author={Misha Stepanov}, journal={Calcolo}, year={2021}, volume={59} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.SIR54","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.SIR54","text":"SIR54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 5th order method suited for SIR-type epidemic models.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Kovalnogov2020RungeKuttaPS, title={Runge–Kutta pairs suited for SIR‐type epidemic models}, author={Vladislav N. Kovalnogov and Theodore E. Simos and Ch. Tsitouras}, journal={Mathematical Methods in the Applied Sciences}, year={2020}, volume={44}, pages={5210 - 5216} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Alshina2","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Alshina2","text":"Alshina2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 2nd order, 2-stage Method with optimal parameters.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Alshina3","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Alshina3","text":"Alshina3(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 3rd order, 3-stage Method with optimal parameters.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowOrderRK/#OrdinaryDiffEqLowOrderRK.Alshina6","page":"OrdinaryDiffEqLowOrderRK","title":"OrdinaryDiffEqLowOrderRK.Alshina6","text":"Alshina6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 6th order, 7-stage Method with optimal parameters.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{Alshina2008, doi = {10.1134/s0965542508030068}, url = {https://doi.org/10.1134/s0965542508030068}, year = {2008}, month = mar, publisher = {Pleiades Publishing Ltd}, volume = {48}, number = {3}, pages = {395–405}, author = {E. A. Alshina and E. M. Zaks and N. N. Kalitkin}, title = {Optimal first- to sixth-order accurate Runge-Kutta schemes}, journal = {Computational Mathematics and Mathematical Physics} }\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"Methods for semi-linear differential equations.","category":"page"},{"location":"semilinear/ExponentialRK/#Installation","page":"OrdinaryDiffEqExponentialRK","title":"Installation","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"To be able to access the solvers in OrdinaryDiffEqLinear, you must first install them use the Julia package manager:","category":"page"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqExponentialRK\")","category":"page"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"semilinear/ExponentialRK/#Example-usage","page":"OrdinaryDiffEqExponentialRK","title":"Example usage","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"using OrdinaryDiffEqExponentialRK, SciMLOperators\nA = [2.0 -1.0; -1.0 2.0]\nlinnonlin_f1 = MatrixOperator(A)\nlinnonlin_f2 = (du, u, p, t) -> du .= 1.01 .* u\nlinnonlin_fun_iip = SplitFunction(linnonlin_f1, linnonlin_f2)\ntspan = (0.0, 1.0)\nu0 = [0.1, 0.1]\nprob = SplitODEProblem(linnonlin_fun_iip, u0, tspan)\nsol = solve(prob, ETDRK4(), dt = 1 / 4)","category":"page"},{"location":"semilinear/ExponentialRK/#Full-list-of-solvers","page":"OrdinaryDiffEqExponentialRK","title":"Full list of solvers","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"LawsonEuler\nNorsettEuler\nETD2\nETDRK2\nETDRK3\nETDRK4\nHochOst4","category":"page"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.LawsonEuler","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.LawsonEuler","text":"LawsonEuler(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver First order exponential Euler scheme.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.NorsettEuler","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.NorsettEuler","text":"NorsettEuler(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver First order exponential-RK scheme. Alias: ETD1\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.ETD2","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.ETD2","text":"ETD2: Exponential Runge-Kutta Method Second order Exponential Time Differencing method (in development).\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.ETDRK2","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.ETDRK2","text":"ETDRK2(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 2nd order exponential-RK scheme.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.ETDRK3","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.ETDRK3","text":"ETDRK3(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 3rd order exponential-RK scheme.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.ETDRK4","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.ETDRK4","text":"ETDRK4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order exponential-RK scheme\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.HochOst4","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.HochOst4","text":"HochOst4(; krylov = false,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order exponential-RK scheme with stiff order 4.\n\nKeyword Arguments\n\nkrylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#Adaptive-Exponential-Rosenbrock-Methods","page":"OrdinaryDiffEqExponentialRK","title":"Adaptive Exponential Rosenbrock Methods","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"Exprb32\nExprb43","category":"page"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.Exprb32","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.Exprb32","text":"Exprb32(; m = 30,\n iop = 0)\n\nSemilinear ODE solver 3rd order adaptive Exponential-Rosenbrock scheme.\n\nKeyword Arguments\n\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.Exprb43","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.Exprb43","text":"Exprb43(; m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order adaptive Exponential-Rosenbrock scheme.\n\nKeyword Arguments\n\nm: Controls the size of Krylov subspace.\niop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#Exponential-Propagation-Iterative-Runge-Kutta-Methods-(EPIRK)","page":"OrdinaryDiffEqExponentialRK","title":"Exponential Propagation Iterative Runge-Kutta Methods (EPIRK)","text":"","category":"section"},{"location":"semilinear/ExponentialRK/","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK","text":"Exp4\nEPIRK4s3A\nEPIRK4s3B\nEPIRK5s3\nEXPRB53s3\nEPIRK5P1\nEPIRK5P2","category":"page"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.Exp4","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.Exp4","text":"Exp4(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order EPIRK scheme.\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nHochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM Journal on Scientific Computing, 19(5), 1552-1574. (https://doi.org/10.1137/S1064827595295337)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EPIRK4s3A","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EPIRK4s3A","text":"EPIRK4s3A(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order EPIRK scheme with stiff order 4.\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nRainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EPIRK4s3B","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EPIRK4s3B","text":"EPIRK4s3B(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 4th order EPIRK scheme with stiff order 4.\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nRainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EPIRK5s3","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EPIRK5s3","text":"EPIRK5s3(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 5th order “horizontal” EPIRK scheme with stiff order 5. Broken.\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nRainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EXPRB53s3","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EXPRB53s3","text":"EXPRB53s3(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 5th order EPIRK scheme with stiff order 5.\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nRainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EPIRK5P1","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EPIRK5P1","text":"EPIRK5P1(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 5th order EPIRK scheme\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nTokman, M., Loffeld, J., & Tranquilli, P. (2012). New Adaptive Exponential Propagation Iterative Methods of Runge–Kutta Type. SIAM Journal on Scientific Computing, 34(5), A2650-A2669. (https://doi.org/10.1137/110849961)\n\n\n\n\n\n","category":"type"},{"location":"semilinear/ExponentialRK/#OrdinaryDiffEqExponentialRK.EPIRK5P2","page":"OrdinaryDiffEqExponentialRK","title":"OrdinaryDiffEqExponentialRK.EPIRK5P2","text":"EPIRK5P2(; adaptive_krylov = true,\n m = 30,\n iop = 0)\n\nSemilinear ODE solver 5th order EPIRK scheme\n\nKeyword Arguments\n\nadaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.\nm: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.\n\nReferences\n\nTokman, M., Loffeld, J., & Tranquilli, P. (2012). New Adaptive Exponential Propagation Iterative Methods of Runge–Kutta Type. SIAM Journal on Scientific Computing, 34(5), A2650-A2669. (https://doi.org/10.1137/110849961)\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"","category":"section"},{"location":"explicit/AdamsBashforthMoulton/","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"Multistep methods, useful for integrating a very expensive to evaluate non-stiff system of differential equations.","category":"page"},{"location":"explicit/AdamsBashforthMoulton/","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqAdamsBashforthMoulton\", \"VCABM\")","category":"page"},{"location":"explicit/AdamsBashforthMoulton/#Full-list-of-solvers","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/AdamsBashforthMoulton/#Explicit-Multistep-Methods","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"Explicit Multistep Methods","text":"","category":"section"},{"location":"explicit/AdamsBashforthMoulton/","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"AB3\nAB4\nAB5","category":"page"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.AB3","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.AB3","text":"AB3()\n\nAdams-Bashforth Explicit Method The 3-step third order multistep method. Ralston's Second Order Method is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.AB4","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.AB4","text":"AB4()\n\nAdams-Bashforth Explicit Method The 4-step fourth order multistep method. Runge-Kutta method of order 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.AB5","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.AB5","text":"AB5()\n\nAdams-Bashforth Explicit Method The 5-step fifth order multistep method. Ralston's 3rd order Runge-Kutta method is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#Predictor-Corrector-Methods","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"Predictor-Corrector Methods","text":"","category":"section"},{"location":"explicit/AdamsBashforthMoulton/","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton","text":"ABM32\nABM43\nABM54\nVCAB3\nVCAB4\nVCAB5\nVCABM3\nVCABM4\nVCABM5\nVCABM","category":"page"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.ABM32","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.ABM32","text":"ABM32()\n\nAdams-Bashforth Explicit Method It is third order method. In ABM32, AB3 works as predictor and Adams Moulton 2-steps method works as Corrector. Ralston's Second Order Method is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.ABM43","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.ABM43","text":"ABM43()\n\nAdams-Bashforth Explicit Method It is fourth order method. In ABM43, AB4 works as predictor and Adams Moulton 3-steps method works as Corrector. Runge-Kutta method of order 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.ABM54","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.ABM54","text":"ABM54()\n\nAdams-Bashforth Explicit Method It is fifth order method. In ABM54, AB5 works as predictor and Adams Moulton 4-steps method works as Corrector. Runge-Kutta method of order 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCAB3","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCAB3","text":"VCAB3()\n\nAdams explicit Method The 3rd order Adams method. Bogacki-Shampine 3/2 method is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCAB4","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCAB4","text":"VCAB4()\n\nAdams explicit Method The 4th order Adams method. Runge-Kutta 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCAB5","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCAB5","text":"VCAB5()\n\nAdams explicit Method The 5th order Adams method. Runge-Kutta 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCABM3","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCABM3","text":"VCABM3()\n\nAdams explicit Method The 3rd order Adams-Moulton method. Bogacki-Shampine 3/2 method is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCABM4","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCABM4","text":"VCABM4()\n\nAdams explicit Method The 4th order Adams-Moulton method. Runge-Kutta 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCABM5","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCABM5","text":"VCABM5()\n\nAdams explicit Method The 5th order Adams-Moulton method. Runge-Kutta 4 is used to calculate starting values.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/AdamsBashforthMoulton/#OrdinaryDiffEqAdamsBashforthMoulton.VCABM","page":"OrdinaryDiffEqAdamsBashforthMoulton","title":"OrdinaryDiffEqAdamsBashforthMoulton.VCABM","text":"VCABM()\n\nadaptive order Adams explicit Method An adaptive order adaptive time Adams Moulton method. It uses an order adaptivity algorithm is derived from Shampine's DDEABM.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Computational Mathematics (2nd revised ed.), Springer (1996) doi: https://doi.org/10.1007/978-3-540-78862-1\n\n\n\n\n\n","category":"type"},{"location":"explicit/QPRK/","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/QPRK/#OrdinaryDiffEqQPRK","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK","text":"","category":"section"},{"location":"explicit/QPRK/","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK","text":"Explicit solvers optimized for a certain number of parallel calls of the system of ordinary differential equations f. Particularly good at low tolerances, when using quad-precision arithmetic, Float128.","category":"page"},{"location":"explicit/QPRK/","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqQPRK\", \"QPRK98\")","category":"page"},{"location":"explicit/QPRK/#Full-list-of-solvers","page":"OrdinaryDiffEqQPRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/QPRK/","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK","text":"QPRK98","category":"page"},{"location":"explicit/QPRK/#OrdinaryDiffEqQPRK.QPRK98","page":"OrdinaryDiffEqQPRK","title":"OrdinaryDiffEqQPRK.QPRK98","text":"QPRK98(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Runge–Kutta pairs of orders 9(8) for use in quadruple precision computations\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nKovalnogov VN, Fedorov RV, Karpukhina TV, Simos TE, Tsitouras C. Runge–Kutta pairs of orders 9 (8) for use in quadruple precision computations. Numerical Algorithms, 2023. doi: https://doi.org/10.1007/s11075-023-01632-8\n\n\n\n\n\n","category":"type"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"implicit/StabalizedIRK/#OrdinaryDiffEqStabalizedIRK","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"","category":"section"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"Stabilized Explicit Runge-Kutta methods, like Runge-Kutta-Chebyshev methods and ROCK methods are explicit methods which chain together many stages in a specific way to get large stability regions. they are made in such a way to converge to a large stability region, and thus suitable to stiff equations. However, they converge to having a large stability region in the direction of the negative real axis, and thus are only stable on a subset of stiff equations which are not dominated by large complex eigenvalues in the Jacobian.","category":"page"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"Stabilized implicit methods try to mitigate this problem by being an IMEX type scheme, requiring a SplitODEProblem where the splitting is designed to treat the large complex eigenvalues implicitly while treating the large real eigenvalues using a fast explicit stabilized RK type of method.","category":"page"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"These methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example","category":"page"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`","category":"page"},{"location":"implicit/StabalizedIRK/#Installation","page":"OrdinaryDiffEqStabalizedIRK","title":"Installation","text":"","category":"section"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"To be able to access the solvers in OrdinaryDiffEqStabalizedIRK, you must first install them use the Julia package manager:","category":"page"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqStabalizedIRK\")","category":"page"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"implicit/StabalizedIRK/#Example-usage","page":"OrdinaryDiffEqStabalizedIRK","title":"Example usage","text":"","category":"section"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"using OrdinaryDiffEqStabilizedIRK\nA = randn(20, 20)\nB = randn(20, 20)\nf1 = (u, p, t) -> A * u\nf2 = (u, p, t) -> B * u\nu0 = randn(20, 1)\ntspan = (0.0, 1.0)\nprob = SplitODEProblem(f1, f2, u0, tspan)\nsol = solve(prob, IRKC())","category":"page"},{"location":"implicit/StabalizedIRK/#Full-list-of-solvers","page":"OrdinaryDiffEqStabalizedIRK","title":"Full list of solvers","text":"","category":"section"},{"location":"implicit/StabalizedIRK/","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabalizedIRK","text":"IRKC","category":"page"},{"location":"implicit/StabalizedIRK/#OrdinaryDiffEqStabilizedIRK.IRKC","page":"OrdinaryDiffEqStabalizedIRK","title":"OrdinaryDiffEqStabilizedIRK.IRKC","text":"IRKC(; eigen_est = nothing)\n\nStabalized Implicit Runge Kutta method. Implicit Runge-Kutta-Chebyshev method.\n\nKeyword Arguments\n\neigen_est: function of the form (integrator) -> integrator.eigen_est = upper_bound, where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.\n\nReferences\n\nREF TBD\n\n\n\n\n\n","category":"type"},{"location":"usage/#Usage","page":"Usage","title":"Usage","text":"","category":"section"},{"location":"usage/","page":"Usage","title":"Usage","text":"OrdinaryDiffEq.jl is part of the SciML common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an OrdinaryDiffEq.jl algorithm to solve. For example, we can solve the ODE tutorial from the docs using the Tsit5() algorithm:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"using OrdinaryDiffEq\nf(u, p, t) = 1.01 * u\nu0 = 1 / 2\ntspan = (0.0, 1.0)\nprob = ODEProblem(f, u0, tspan)\nsol = solve(prob, Tsit5(), reltol = 1e-8, abstol = 1e-8)\nusing Plots\nplot(sol, linewidth = 5, title = \"Solution to the linear ODE with a thick line\",\n xaxis = \"Time (t)\", yaxis = \"u(t) (in μm)\", label = \"My Thick Line!\") # legend=false\nplot!(sol.t, t -> 0.5 * exp(1.01t), lw = 3, ls = :dash, label = \"True Solution!\")","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"That example uses the out-of-place syntax f(u,p,t), while the inplace syntax (more efficient for systems of equations) is shown in the Lorenz example:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"using OrdinaryDiffEq\nfunction lorenz(du, u, p, t)\n du[1] = 10.0(u[2] - u[1])\n du[2] = u[1] * (28.0 - u[3]) - u[2]\n du[3] = u[1] * u[2] - (8 / 3) * u[3]\nend\nu0 = [1.0; 0.0; 0.0]\ntspan = (0.0, 100.0)\nprob = ODEProblem(lorenz, u0, tspan)\nsol = solve(prob, Tsit5())\nusing Plots;\nplot(sol, vars = (1, 2, 3));","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"Very fast static array versions can be specifically compiled to the size of your model. For example:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"using OrdinaryDiffEq, StaticArrays\nfunction lorenz(u, p, t)\n SA[10.0(u[2] - u[1]), u[1] * (28.0 - u[3]) - u[2], u[1] * u[2] - (8 / 3) * u[3]]\nend\nu0 = SA[1.0; 0.0; 0.0]\ntspan = (0.0, 100.0)\nprob = ODEProblem(lorenz, u0, tspan)\nsol = solve(prob, Tsit5())","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"For “refined ODEs”, like dynamical equations and SecondOrderODEProblems, refer to the DiffEqDocs. For example, in DiffEqTutorials.jl we show how to solve equations of motion using symplectic methods:","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"function HH_acceleration(dv, v, u, p, t)\n x, y = u\n dx, dy = dv\n dv[1] = -x - 2x * y\n dv[2] = y^2 - y - x^2\nend\ninitial_positions = [0.0, 0.1]\ninitial_velocities = [0.5, 0.0]\nprob = SecondOrderODEProblem(HH_acceleration, initial_velocities, initial_positions, tspan)\nsol2 = solve(prob, KahanLi8(), dt = 1 / 10);","category":"page"},{"location":"usage/","page":"Usage","title":"Usage","text":"Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).","category":"page"},{"location":"usage/#Available-Solvers","page":"Usage","title":"Available Solvers","text":"","category":"section"},{"location":"usage/","page":"Usage","title":"Usage","text":"For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.","category":"page"},{"location":"explicit/Feagin/","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/Feagin/#OrdinaryDiffEqFeagin","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin","text":"","category":"section"},{"location":"explicit/Feagin/","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin","text":"Preferred solvers for non-stiff problems at very low tolerance, <1e-30. Best combined with preciser than Float64 number types for the state, such as the BigFloat number type. Note that the Feagin methods have a less robust error estimator than the Verner methods, and thus even for very low tolerance problems the Verner methods (Vern9) may still be more efficient. In addition, at extremely low tolerances the explicit extrapolation methods allow for arbitrarily high variable order stepping which will also outperform the Feagin methods. As such, the Feagin methods may be useful in the Float128 precision range but should be tested against other algorithms.","category":"page"},{"location":"explicit/Feagin/","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqFeagin\", \"Feagin14\")","category":"page"},{"location":"explicit/Feagin/#Full-list-of-solvers","page":"OrdinaryDiffEqFeagin","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/Feagin/","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin","text":"Feagin10\nFeagin12\nFeagin14","category":"page"},{"location":"explicit/Feagin/#OrdinaryDiffEqFeagin.Feagin10","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin.Feagin10","text":"Feagin10(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nExplicit Runge-Kutta Method. Feagin's 10th-order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \\& Scientific Computations} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Feagin/#OrdinaryDiffEqFeagin.Feagin12","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin.Feagin12","text":"Feagin12(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nExplicit Runge-Kutta Method. Feagin's 12th-order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \\& Scientific Computations} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Feagin/#OrdinaryDiffEqFeagin.Feagin14","page":"OrdinaryDiffEqFeagin","title":"OrdinaryDiffEqFeagin.Feagin14","text":"Feagin14(; step_limiter! = OrdinaryDiffEq.trivial_limiter!)\n\nExplicit Runge-Kutta Method. Feagin's 14th-order method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\n\nReferences\n\n@article{feagin2009explicit, title={An Explicit Runge-Kutta Method of Order Fourteen}, author={Feagin, Terry}, year={2009}, publisher={Numerical Algorithms} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/Extrapolation/#OrdinaryDiffEqExtrapolation","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"","category":"section"},{"location":"explicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"Solvers based on within method parallelism, allowing multithreading of the solution across different values of f. The explicit extrapolation solvers are generally outclassed by other explicit methods. However, some stiff extrapolation methods perform very well if the problem is sufficiently stiff.","category":"page"},{"location":"explicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqExtrapolation\", \"ExtrapolationMidpointDeuflhard\")","category":"page"},{"location":"explicit/Extrapolation/#Full-list-of-solvers","page":"OrdinaryDiffEqExtrapolation","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/Extrapolation/","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation","text":"AitkenNeville\nExtrapolationMidpointDeuflhard\nExtrapolationMidpointHairerWanner","category":"page"},{"location":"explicit/Extrapolation/#OrdinaryDiffEqExtrapolation.AitkenNeville","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.AitkenNeville","text":"AitkenNeville(; max_order::Int = 10,\n min_order::Int = 1,\n init_order = 3,\n thread = OrdinaryDiffEq.False())\n\nParallelized Explicit Extrapolation Method. Euler extrapolation using Aitken-Neville with the Romberg Sequence.\n\nKeyword Arguments\n\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ExtrapolationMidpointDeuflhard","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ExtrapolationMidpointDeuflhard","text":"ExtrapolationMidpointDeuflhard(; max_order = 10,\n min_order = 1,\n init_order = 5,\n thread = OrdinaryDiffEq.True(),\n sequence = :harmonic,\n sequence_factor = 2)\n\nParallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates.\n\nKeyword Arguments\n\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\nsequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/Extrapolation/#OrdinaryDiffEqExtrapolation.ExtrapolationMidpointHairerWanner","page":"OrdinaryDiffEqExtrapolation","title":"OrdinaryDiffEqExtrapolation.ExtrapolationMidpointHairerWanner","text":"ExtrapolationMidpointHairerWanner(; max_order = 10,\n min_order = 2,\n init_order = 5,\n thread = OrdinaryDiffEq.True(),\n sequence = :harmonic,\n sequence_factor = 2)\n\nParallelized Explicit Extrapolation Method. Midpoint extrapolation using Barycentric coordinates, following Hairer's ODEX in the adaptivity behavior.\n\nKeyword Arguments\n\nmax_order: maximum order of the adaptive order algorithm.\nmin_order: minimum order of the adaptive order algorithm.\ninit_order: initial order of the adaptive order algorithm.\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nsequence: the step-number sequences, also called the subdividing sequence. Possible values are :harmonic, :romberg or :bulirsch.\nsequence_factor: denotes which even multiple of sequence to take while evaluating internal discretizations.\n\nReferences\n\n@inproceedings{elrod2022parallelizing, title={Parallelizing explicit and implicit extrapolation methods for ordinary differential equations}, author={Elrod, Chris and Ma, Yingbo and Althaus, Konstantin and Rackauckas, Christopher and others}, booktitle={2022 IEEE High Performance Extreme Computing Conference (HPEC)}, pages={1–9}, year={2022}, organization={IEEE}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK","text":"","category":"section"},{"location":"explicit/LowStorageRK/","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK","text":"These methods are designed to have reduced register requirements, allowing for larger sets of ODEs to more easily fit into RAM. For example, while the 5th order Tsit5 requires around 9 concurrent instantiations of the ODE state u, RDPK3Sp510 can achieve 5th order with 3 registers, effectively requiring 1/3 of the memory. However, there are some efficiency trade-offs used in the design of the low-storage RK methods, and thus they are generally only recommended in situations which are RAM bound, like large-scale PDE discretizations.","category":"page"},{"location":"explicit/LowStorageRK/","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqLowStorageRK\", \"CarpenterKennedy2N54\")","category":"page"},{"location":"explicit/LowStorageRK/#Full-list-of-solvers","page":"OrdinaryDiffEqLowStorageRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/LowStorageRK/","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK","text":"ORK256\nDGLDDRK73_C\nCarpenterKennedy2N54\nNDBLSRK124\nNDBLSRK144\nCFRLDDRK64\nTSLDDRK74\nDGLDDRK84_C\nDGLDDRK84_F\nSHLDDRK64\nRK46NL\nParsaniKetchesonDeconinck3S32\nParsaniKetchesonDeconinck3S82\nParsaniKetchesonDeconinck3S53\nParsaniKetchesonDeconinck3S173\nParsaniKetchesonDeconinck3S94\nParsaniKetchesonDeconinck3S184\nParsaniKetchesonDeconinck3S105\nParsaniKetchesonDeconinck3S205\nCKLLSRK43_2\nCKLLSRK54_3C\nCKLLSRK95_4S\nCKLLSRK95_4C\nCKLLSRK95_4M\nCKLLSRK54_3C_3R\nCKLLSRK54_3M_3R\nCKLLSRK54_3N_3R\nCKLLSRK85_4C_3R\nCKLLSRK85_4M_3R\nCKLLSRK85_4P_3R\nCKLLSRK54_3N_4R\nCKLLSRK54_3M_4R\nCKLLSRK65_4M_4R\nCKLLSRK85_4FM_4R\nCKLLSRK75_4M_5R\nRDPK3Sp35\nRDPK3SpFSAL35\nRDPK3Sp49\nRDPK3SpFSAL49\nRDPK3Sp510\nRDPK3SpFSAL510\nHSLDDRK64\nNDBLSRK134\nSHLDDRK_2N\nSHLDDRK52","category":"page"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ORK256","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ORK256","text":"ORK256(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. A second-order, five-stage method for wave propagation equations. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nMatteo Bernardini, Sergio Pirozzoli. A General Strategy for the Optimization of Runge-Kutta Schemes for Wave Propagation Phenomena. Journal of Computational Physics, 228(11), pp 4182-4199, 2009. doi: https://doi.org/10.1016/j.jcp.2009.02.032\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.DGLDDRK73_C","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.DGLDDRK73_C","text":"DGLDDRK73_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 7-stage, third order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nT. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54","text":"CarpenterKennedy2N54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. A fourth-order, five-stage low-storage method of Carpenter and Kennedy (free 3rd order Hermite interpolant). Fixed timestep only. Designed for hyperbolic PDEs (stability properties).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\n@article{carpenter1994fourth, title={Fourth-order 2N-storage Runge-Kutta schemes}, author={Carpenter, Mark H and Kennedy, Christopher A}, year={1994} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.NDBLSRK124","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.NDBLSRK124","text":"NDBLSRK124(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 12-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nJens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.NDBLSRK144","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.NDBLSRK144","text":"NDBLSRK144(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 14-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nJens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CFRLDDRK64","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CFRLDDRK64","text":"CFRLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-storage, low-dissipation, low-dispersion scheme. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nM. Calvo, J. M. Franco, L. Randez. A New Minimum Storage Runge–Kutta Scheme for Computational Acoustics. Journal of Computational Physics, 201, pp 1-12, 2004. doi: https://doi.org/10.1016/j.jcp.2004.05.012\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.TSLDDRK74","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.TSLDDRK74","text":"TSLDDRK74(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 7-stage, fourth order low-storage low-dissipation, low-dispersion scheme with maximal accuracy and stability limit along the imaginary axes. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nKostas Tselios, T. E. Simos. Optimized Runge–Kutta Methods with Minimal Dispersion and Dissipation for Problems arising from Computational Acoustics. Physics Letters A, 393(1-2), pp 38-47, 2007. doi: https://doi.org/10.1016/j.physleta.2006.10.072\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.DGLDDRK84_C","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.DGLDDRK84_C","text":"DGLDDRK84_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when maximum spatial step is small due to geometric features of computational domain. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nT. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.DGLDDRK84_F","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.DGLDDRK84_F","text":"DGLDDRK84_F(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 8-stage, fourth order low-storage low-dissipation, low-dispersion scheme for discontinuous Galerkin space discretizations applied to wave propagation problems. Optimized for PDE discretizations when the maximum spatial step size is not constrained. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nT. Toulorge, W. Desmet. Optimal Runge–Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems. Journal of Computational Physics, 231(4), pp 2067-2091, 2012. doi: https://doi.org/10.1016/j.jcp.2011.11.024\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.SHLDDRK64","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.SHLDDRK64","text":"SHLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. A fourth-order, six-stage low-storage method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nD. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RK46NL","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RK46NL","text":"RK46NL(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nJulien Berland, Christophe Bogey, Christophe Bailly. Low-Dissipation and Low-Dispersion Fourth-Order Runge-Kutta Algorithm. Computers & Fluids, 35(10), pp 1459-1463, 2006. doi: https://doi.org/10.1016/j.compfluid.2005.04.003\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32","text":"ParsaniKetchesonDeconinck3S32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 3-stage, second order (3S) low-storage scheme, optimized the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82","text":"ParsaniKetchesonDeconinck3S82(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 8-stage, second order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53","text":"ParsaniKetchesonDeconinck3S53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173","text":"ParsaniKetchesonDeconinck3S173(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 17-stage, third order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94","text":"ParsaniKetchesonDeconinck3S94(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 9-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184","text":"ParsaniKetchesonDeconinck3S184(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 18-stage, fourth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105","text":"ParsaniKetchesonDeconinck3S105(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 10-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205","text":"ParsaniKetchesonDeconinck3S205(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 20-stage, fifth order (3S) low-storage scheme, optimized for the spectral difference method applied to wave propagation problems.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nParsani, Matteo, David I. Ketcheson, and W. Deconinck. Optimized explicit Runge–Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing 35.2 (2013): A957-A986. doi: https://doi.org/10.1137/120885899\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK43_2","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK43_2","text":"CKLLSRK43_2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 4-stage, third order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C","text":"CKLLSRK54_3C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S","text":"CKLLSRK95_4S(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C","text":"CKLLSRK95_4C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M","text":"CKLLSRK95_4M(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R","text":"CKLLSRK54_3C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R","text":"CKLLSRK54_3M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R","text":"CKLLSRK54_3N_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R","text":"CKLLSRK85_4C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R","text":"CKLLSRK85_4M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R","text":"CKLLSRK85_4P_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R","text":"CKLLSRK54_3N_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R","text":"CKLLSRK54_3M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R","text":"CKLLSRK65_4M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. 6-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R","text":"CKLLSRK85_4FM_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R","text":"CKLLSRK75_4M_5R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. CKLLSRK754M5R: Low-Storage Method 7-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kennedy2000low, title={Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations}, author={Kennedy, Christopher A and Carpenter, Mark H and Lewis, R Michael}, journal={Applied numerical mathematics}, volume={35}, number={3}, pages={177–219}, year={2000}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3Sp35","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3Sp35","text":"RDPK3Sp35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35","text":"RDPK3SpFSAL35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3Sp49","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3Sp49","text":"RDPK3Sp49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fourth-order, nine-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49","text":"RDPK3SpFSAL49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fourth-order, nine-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3Sp510","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3Sp510","text":"RDPK3Sp510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fifth-order, ten-stage method with embedded error estimator designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510","text":"RDPK3SpFSAL510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fifth-order, ten-stage method with embedded error estimator using the FSAL property designed for spectral element discretizations of compressible fluid mechanics.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRanocha, Dalcin, Parsani, Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.HSLDDRK64","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.HSLDDRK64","text":"HSLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. Low-Storage Method 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nD. Stanescu, W. G. Habashi. 2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics. Journal of Computational Physics, 143(2), pp 674-681, 1998. doi: https://doi.org/10.1006/jcph.1998.5986 }\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.NDBLSRK134","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.NDBLSRK134","text":"NDBLSRK134(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n williamson_condition = true)\n\nExplicit Runge-Kutta Method. 13-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nwilliamson_condition: allows for an optimization that allows fusing broadcast expressions with the function call f. However, it only works for Array types.\n\nReferences\n\nJens Niegemann, Richard Diehl, Kurt Busch. Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions. Journal of Computational Physics, 231, pp 364-372, 2012. doi: https://doi.org/10.1016/j.jcp.2011.09.003\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.SHLDDRK_2N","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.SHLDDRK_2N","text":"SHLDDRK_2N(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low dissipation and dispersion Runge-Kutta schemes for computational acoustics\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{stanescu19982n, title={2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics}, author={Stanescu, D and Habashi, WG}, journal={Journal of Computational Physics}, volume={143}, number={2}, pages={674–681}, year={1998}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/LowStorageRK/#OrdinaryDiffEqLowStorageRK.SHLDDRK52","page":"OrdinaryDiffEqLowStorageRK","title":"OrdinaryDiffEqLowStorageRK.SHLDDRK52","text":"SHLDDRK52(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Low dissipation and dispersion Runge-Kutta schemes for computational acoustics\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{stanescu19982n, title={2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics}, author={Stanescu, D and Habashi, WG}, journal={Journal of Computational Physics}, volume={143}, number={2}, pages={674–681}, year={1998}, publisher={Elsevier}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/Verner/","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"","category":"section"},{"location":"explicit/Verner/","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"Preferred solvers for non-stiff problems at low tolerance. Vern6, Vern7, or Vern8 are good methods for tolerances between ~1e-8-1e-12, and using Float64 numbers for the state of the differential equation. For even lower tolerances,Vern9 should be used, combined with the more precise BigFloat number type.","category":"page"},{"location":"explicit/Verner/","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqVerner\", \"Vern6\")","category":"page"},{"location":"explicit/Verner/#Full-list-of-solvers","page":"OrdinaryDiffEqVerner","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/Verner/","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"Vern6\nVern7\nVern8\nVern9","category":"page"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.Vern6","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.Vern6","text":"Vern6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n lazy = true)\n\nExplicit Runge-Kutta Method. Verner's “Most Efficient” 6/5 Runge-Kutta method. (lazy 6th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nlazy: determines if the lazy interpolant is used.\n\nReferences\n\n@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.Vern7","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.Vern7","text":"Vern7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n lazy = true)\n\nExplicit Runge-Kutta Method. Verner's “Most Efficient” 7/6 Runge-Kutta method. (lazy 7th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nlazy: determines if the lazy interpolant is used.\n\nReferences\n\n@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.Vern8","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.Vern8","text":"Vern8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n lazy = true)\n\nExplicit Runge-Kutta Method. Verner's “Most Efficient” 8/7 Runge-Kutta method. (lazy 8th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nlazy: determines if the lazy interpolant is used.\n\nReferences\n\n@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.Vern9","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.Vern9","text":"Vern9(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False(),\n lazy = true)\n\nExplicit Runge-Kutta Method. Verner's “Most Efficient” 9/8 Runge-Kutta method. (lazy9th order interpolant).\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\nlazy: determines if the lazy interpolant is used.\n\nReferences\n\n@article{verner2010numerically, title={Numerically optimal Runge–Kutta pairs with interpolants}, author={Verner, James H}, journal={Numerical Algorithms}, volume={53}, number={2-3}, pages={383–396}, year={2010}, publisher={Springer} }\n\n\n\n\n\n","category":"type"},{"location":"explicit/Verner/","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner","text":"AutoVern6\nAutoVern7\nAutoVern8\nAutoVern9","category":"page"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.AutoVern6","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.AutoVern6","text":"Automatic switching algorithm that can switch between the (non-stiff) Vern6() and stiff_alg.\n\nAutoVern6(stiff_alg; kwargs...)\n\nThis method is equivalent to AutoAlgSwitch(Vern6(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.\n\n\n\n\n\n","category":"function"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.AutoVern7","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.AutoVern7","text":"Automatic switching algorithm that can switch between the (non-stiff) Vern7() and stiff_alg.\n\nAutoVern7(stiff_alg; kwargs...)\n\nThis method is equivalent to AutoAlgSwitch(Vern7(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.\n\n\n\n\n\n","category":"function"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.AutoVern8","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.AutoVern8","text":"Automatic switching algorithm that can switch between the (non-stiff) Vern8() and stiff_alg.\n\nAutoVern8(stiff_alg; kwargs...)\n\nThis method is equivalent to AutoAlgSwitch(Vern8(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.\n\n\n\n\n\n","category":"function"},{"location":"explicit/Verner/#OrdinaryDiffEqVerner.AutoVern9","page":"OrdinaryDiffEqVerner","title":"OrdinaryDiffEqVerner.AutoVern9","text":"Automatic switching algorithm that can switch between the (non-stiff) Vern9() and stiff_alg.\n\nAutoVern9(stiff_alg; kwargs...)\n\nThis method is equivalent to AutoAlgSwitch(Vern9(), stiff_alg; kwargs...). To gain access to stiff algorithms you might have to install additional libraries, such as OrdinaryDiffEqRosenbrock.\n\n\n\n\n\n","category":"function"},{"location":"#OrdinaryDiffEq.jl","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.","category":"page"},{"location":"#Installation","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"Installation","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"import Pkg;\nPkg.add(\"OrdinaryDiffEq\");","category":"page"},{"location":"#Reproducibility","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"
","category":"page"},{"location":"","page":"OrdinaryDiffEq.jl: ODE solvers and utilities","title":"OrdinaryDiffEq.jl: ODE solvers and utilities","text":"using TOML\nusing Markdown\nversion = TOML.parse(read(\"../../Project.toml\", String))[\"version\"]\nname = TOML.parse(read(\"../../Project.toml\", String))[\"name\"]\nlink_manifest = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Manifest.toml\"\nlink_project = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Project.toml\"\nMarkdown.parse(\"\"\"You can also download the\n[manifest]($link_manifest)\nfile and the\n[project]($link_project)\nfile.\n\"\"\")","category":"page"},{"location":"explicit/SSPRK/","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"","category":"section"},{"location":"explicit/SSPRK/","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"SSPRK methods are Runge-Kutta methods which support the \"strongly preserving property\" (SSP). They are designed for the use in discretizations of hyperbolic partial differential equations and conservation laws as they have extra stability properties ( e.g., stability with respect to total variation, the maximum norm, or other convex functionals) when step-size restrictions are respected. In particular, these properties are granted if the step-size is kept to a level where the CFL coefficients are less than the SSP coefficient.","category":"page"},{"location":"explicit/SSPRK/","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"Note that for SSPRK methods, a algorithm utility OrdinaryDiffEqCore.ssp_coefficient(alg) exists that allows for querying the SSP coefficient for use in step size calculations.","category":"page"},{"location":"explicit/SSPRK/","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"first_steps = evalfile(\"./common_first_steps.jl\")\nfirst_steps(\"OrdinaryDiffEqSSPRK\", \"SSPRK22\")","category":"page"},{"location":"explicit/SSPRK/#Full-list-of-solvers","page":"OrdinaryDiffEqSSPRK","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/SSPRK/","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK","text":"SSPRK22\nSSPRK33\nSSPRK53\nKYKSSPRK42\nKYK2014DGSSPRK_3S2\nSSPRK53_2N1\nSSPRK53_2N2\nSSPRK53_H\nSSPRK63\nSSPRK73\nSSPRK83\nSSPRK43\nSSPRK432\nSSPRKMSVS43\nSSPRKMSVS32\nSSPRK932\nSSPRK54\nSSPRK104","category":"page"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK22","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK22","text":"SSPRK22(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A second-order, two-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nShu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK33","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK33","text":"SSPRK33(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, three-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nShu, Chi-Wang, and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77.2 (1988): 439-471. https://doi.org/10.1016/0021-9991(88)90177-5\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK53","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK53","text":"SSPRK53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRuuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.KYKSSPRK42","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.KYKSSPRK42","text":"KYKSSPRK42(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kubatko2014optimal, title={Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods}, author={Kubatko, Ethan J and Yeager, Benjamin A and Ketcheson, David I}, journal={Journal of Scientific Computing}, volume={60}, pages={313–344}, year={2014}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.KYK2014DGSSPRK_3S2","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.KYK2014DGSSPRK_3S2","text":"KYK2014DGSSPRK_3S2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\n@article{kubatko2014optimal, title={Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods}, author={Kubatko, Ethan J and Yeager, Benjamin A and Ketcheson, David I}, journal={Journal of Scientific Computing}, volume={60}, pages={313–344}, year={2014}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK53_2N1","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK53_2N1","text":"SSPRK53_2N1(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nHigueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK53_2N2","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK53_2N2","text":"SSPRK53_2N2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nHigueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK53_H","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK53_H","text":"SSPRK53_H(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nHigueras and T. Roldán. New third order low-storage SSP explicit Runge–Kutta methods arXiv:1809.04807v1.\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK63","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK63","text":"SSPRK63(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, six-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRuuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK73","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK73","text":"SSPRK73(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, seven-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRuuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK83","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK83","text":"SSPRK83(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, eight-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRuuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK43","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK43","text":"SSPRK43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nOptimal third-order explicit SSP method with four stages discovered by\n\nJ. F. B. M. Kraaijevanger. \"Contractivity of Runge-Kutta methods.\" In: BIT Numerical Mathematics 31.3 (1991), pp. 482–528. DOI: 10.1007/BF01933264.\n\nEmbedded method constructed by\n\nSidafa Conde, Imre Fekete, John N. Shadid. \"Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods.\" arXiv: 1806.08693\n\nEfficient implementation (and optimized controller) developed by\n\nHendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics arXiv:2104.06836\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK432","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK432","text":"SSPRK432(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, four-stage explicit strong stability preserving (SSP) method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nGottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011. Example 6.1\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRKMSVS43","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRKMSVS43","text":"SSPRKMSVS43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, four-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nShu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRKMSVS32","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRKMSVS32","text":"SSPRKMSVS32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A second-order, three-step explicit strong stability preserving (SSP) linear multistep method. This method does not come with an error estimator and requires a fixed time step size.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nShu, Chi-Wang. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084. DOI: 10.1137/0909073\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK932","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK932","text":"SSPRK932(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A third-order, nine-stage explicit strong stability preserving (SSP) method.\n\nConsider using SSPRK43 instead, which uses the same main method and an improved embedded method.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nGottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011.\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK54","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK54","text":"SSPRK54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fourth-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nRuuth, Steven. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75.253 (2006): 183-207.\n\n\n\n\n\n","category":"type"},{"location":"explicit/SSPRK/#OrdinaryDiffEqSSPRK.SSPRK104","page":"OrdinaryDiffEqSSPRK","title":"OrdinaryDiffEqSSPRK.SSPRK104","text":"SSPRK104(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,\n step_limiter! = OrdinaryDiffEq.trivial_limiter!,\n thread = OrdinaryDiffEq.False())\n\nExplicit Runge-Kutta Method. A fourth-order, ten-stage explicit strong stability preserving (SSP) method. Fixed timestep only.\n\nKeyword Arguments\n\nstage_limiter!: function of the form limiter!(u, integrator, p, t)\nstep_limiter!: function of the form limiter!(u, integrator, p, t)\nthread: determines whether internal broadcasting on appropriate CPU arrays should be serial (thread = OrdinaryDiffEq.False()) or use multiple threads (thread = OrdinaryDiffEq.True()) when Julia is started with multiple threads.\n\nReferences\n\nKetcheson, David I. Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM Journal on Scientific Computing 30.4 (2008): 2113-2136.\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"CollapsedDocStrings = true","category":"page"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"","category":"section"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"Second order solvers.","category":"page"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"To be able to access the solvers in OrdinaryDiffEqRKN, you must first install them use the Julia package manager:","category":"page"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"using Pkg\nPkg.add(\"OrdinaryDiffEqRKN\")","category":"page"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.","category":"page"},{"location":"explicit/RKN/#Example-usage","page":"OrdinaryDiffEqRKN","title":"Example usage","text":"","category":"section"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"using OrdinaryDiffEqOrdinaryDiffEqRKN\nfunction HH_acceleration!(dv, v, u, p, t)\n x, y = u\n dx, dy = dv\n dv[1] = -x - 2x * y\n dv[2] = y^2 - y - x^2\nend\ninitial_positions = [0.0, 0.1]\ninitial_velocities = [0.5, 0.0]\ntspan = (0.0, 1.0)\nprob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)\nsol = solve(prob, Nystrom4(), dt = 1 / 10)","category":"page"},{"location":"explicit/RKN/#Full-list-of-solvers","page":"OrdinaryDiffEqRKN","title":"Full list of solvers","text":"","category":"section"},{"location":"explicit/RKN/","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN","text":"IRKN3\nIRKN4\nNystrom4\nNystrom4VelocityIndependent\nNystrom5VelocityIndependent\nFineRKN4\nFineRKN5\nDPRKN4\nDPRKN5\nDPRKN6\nDPRKN6FM\nDPRKN8\nDPRKN12\nERKN4\nERKN5\nERKN7\nRKN4","category":"page"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.IRKN3","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.IRKN3","text":"IRKN3()\n\nImproved Runge-Kutta-Nyström method Method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not depend on the first derivative.\n\nKeyword Arguments\n\nReferences\n\n@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.IRKN4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.IRKN4","text":"IRKN4()\n\nImproved Runge-Kutta-Nyström method Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not be dependent on the first derivative. Recommended for smooth problems with expensive functions to evaluate.\n\nKeyword Arguments\n\nReferences\n\n@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.Nystrom4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.Nystrom4","text":"Nystrom4()\n\nImproved Runge-Kutta-Nyström method A 4th order explicit method which can be applied directly on second order ODEs. Can only be used with fixed time steps. In case the ODE Problem is not dependent on the first derivative consider using Nystrom4VelocityIndependent to increase performance.\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.Nystrom4VelocityIndependent","text":"Nystrom4VelocityIndependent()\n\nImproved Runge-Kutta-Nyström method A 4th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.Nystrom5VelocityIndependent","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.Nystrom5VelocityIndependent","text":"Nystrom5VelocityIndependent()\n\nImproved Runge-Kutta-Nyström method A 5th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).\n\nKeyword Arguments\n\nReferences\n\nE. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.FineRKN4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.FineRKN4","text":"FineRKN4()\n\nImproved Runge-Kutta-Nyström method A 4th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.\n\nKeyword Arguments\n\nReferences\n\n@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{\"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.FineRKN5","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.FineRKN5","text":"FineRKN5()\n\nImproved Runge-Kutta-Nyström method A 5th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.\n\nKeyword Arguments\n\nReferences\n\n@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{\"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN4","text":"DPRKN4()\n\nImproved Runge-Kutta-Nyström method 4th order explicit method. The second order ODE should not depend on the first derivative.\n\nKeyword Arguments\n\nReferences\n\n@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN5","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN5","text":"DPRKN5()\n\nImproved Runge-Kutta-Nyström method 5th order explicit method. The second order ODE should not depend on the first derivative.\n\nKeyword Arguments\n\nReferences\n\n@article{Bettis1973ARN, title={A Runge-Kutta Nystrom algorithm}, author={Dale G. Bettis}, journal={Celestial mechanics}, year={1973}, volume={8}, pages={229-233}, publisher={Springer}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN6","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN6","text":"DPRKN6()\n\nImproved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Free 6th order interpolant\n\nKeyword Arguments\n\nReferences\n\n@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN6FM","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN6FM","text":"DPRKN6FM()\n\nImproved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Compared to DPRKN6, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.\n\nKeyword Arguments\n\nReferences\n\n@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN8","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN8","text":"DPRKN8()\n\nImproved Runge-Kutta-Nyström method 8th order explicit method. The second order ODE should not depend on the first derivative. Not as efficient as DPRKN12 when high accuracy is needed, however this solver is competitive with DPRKN6 at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.\n\nKeyword Arguments\n\nReferences\n\n@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.DPRKN12","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.DPRKN12","text":"DPRKN12()\n\nImproved Runge-Kutta-Nyström method 12th order explicit method. The second order ODE should not depend on the first derivative. Most efficient when high accuracy is needed.\n\nKeyword Arguments\n\nReferences\n\n@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.ERKN4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.ERKN4","text":"ERKN4()\n\nImproved Runge-Kutta-Nyström method Embedded 4(3) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.\n\nKeyword Arguments\n\nReferences\n\n@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{\"o}m Method for Solving Periodic Initial Value Problems}, author={Demba, MA and Senu, N and Ismail, F}, journal={Applied Mathematical Sciences}, volume={11}, number={17}, pages={819–838}, year={2017}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.ERKN5","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.ERKN5","text":"ERKN5()\n\nImproved Runge-Kutta-Nyström method Embedded 5(4) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.\n\nKeyword Arguments\n\nReferences\n\n@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{\"o}m Methods for the Numerical Solution of Oscillatory Initial Value Problems}, author={Demba, Musa A and Senu, Norazak and Ismail, Fudziah}, journal={Mathematical and Computational Applications}, volume={21}, number={4}, pages={46}, year={2016}, publisher={Multidisciplinary Digital Publishing Institute}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.ERKN7","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.ERKN7","text":"ERKN7()\n\nImproved Runge-Kutta-Nyström method Embedded pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.\n\nKeyword Arguments\n\nReferences\n\n@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{\"o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753}}\n\n\n\n\n\n","category":"type"},{"location":"explicit/RKN/#OrdinaryDiffEqRKN.RKN4","page":"OrdinaryDiffEqRKN","title":"OrdinaryDiffEqRKN.RKN4","text":"RKN4()\n\nImproved Runge-Kutta-Nyström method 3 stage fourth order method to solve second order linear inhomogeneous IVPs. Does not include an adaptive method. Solves for for d-dimensional differential systems of second order linear inhomogeneous equations.\n\nwarning: Warning\n\n\nThis method is only fourth order for these systems, the method is second order otherwise!\n\nKeyword Arguments\n\nReferences\n\n@article{MONTIJANO2024115533, title = {Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs}, author = {J.I. Montijano and L. Rández and M. Calvo}, journal = {Journal of Computational and Applied Mathematics}, volume = {438}, pages = {115533}, year = {2024},}\n\n\n\n\n\n","category":"type"}] } diff --git a/dev/semilinear/ExponentialRK/index.html b/dev/semilinear/ExponentialRK/index.html new file mode 100644 index 0000000000..1ef230014d --- /dev/null +++ b/dev/semilinear/ExponentialRK/index.html @@ -0,0 +1,43 @@ + +OrdinaryDiffEqExponentialRK · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqExponentialRK

Methods for semi-linear differential equations.

Installation

To be able to access the solvers in OrdinaryDiffEqLinear, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqExponentialRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqExponentialRK, SciMLOperators
+A = [2.0 -1.0; -1.0 2.0]
+linnonlin_f1 = MatrixOperator(A)
+linnonlin_f2 = (du, u, p, t) -> du .= 1.01 .* u
+linnonlin_fun_iip = SplitFunction(linnonlin_f1, linnonlin_f2)
+tspan = (0.0, 1.0)
+u0 = [0.1, 0.1]
+prob = SplitODEProblem(linnonlin_fun_iip, u0, tspan)
+sol = solve(prob, ETDRK4(), dt = 1 / 4)

Full list of solvers

OrdinaryDiffEqExponentialRK.LawsonEuler — Type
LawsonEuler(; krylov = false,
+              m = 30,
+              iop = 0)

Semilinear ODE solver First order exponential Euler scheme.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source
OrdinaryDiffEqExponentialRK.NorsettEuler — Type
NorsettEuler(; krylov = false,
+               m = 30,
+               iop = 0)

Semilinear ODE solver First order exponential-RK scheme. Alias: ETD1

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source
OrdinaryDiffEqExponentialRK.ETDRK2 — Type
ETDRK2(; krylov = false,
+         m = 30,
+         iop = 0)

Semilinear ODE solver 2nd order exponential-RK scheme.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source
OrdinaryDiffEqExponentialRK.ETDRK3 — Type
ETDRK3(; krylov = false,
+         m = 30,
+         iop = 0)

Semilinear ODE solver 3rd order exponential-RK scheme.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source
OrdinaryDiffEqExponentialRK.ETDRK4 — Type
ETDRK4(; krylov = false,
+         m = 30,
+         iop = 0)

Semilinear ODE solver 4th order exponential-RK scheme

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source
OrdinaryDiffEqExponentialRK.HochOst4 — Type
HochOst4(; krylov = false,
+           m = 30,
+           iop = 0)

Semilinear ODE solver 4th order exponential-RK scheme with stiff order 4.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

Adaptive Exponential Rosenbrock Methods

OrdinaryDiffEqExponentialRK.Exprb32 — Type
Exprb32(; m = 30,
+          iop = 0)

Semilinear ODE solver 3rd order adaptive Exponential-Rosenbrock scheme.

Keyword Arguments

  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)

source
OrdinaryDiffEqExponentialRK.Exprb43 — Type
Exprb43(; m = 30,
+          iop = 0)

Semilinear ODE solver 4th order adaptive Exponential-Rosenbrock scheme.

Keyword Arguments

  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)

source

Exponential Propagation Iterative Runge-Kutta Methods (EPIRK)

OrdinaryDiffEqExponentialRK.Exp4 — Type
Exp4(; adaptive_krylov = true,
+       m = 30,
+       iop = 0)

Semilinear ODE solver 4th order EPIRK scheme.

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM Journal on Scientific Computing, 19(5), 1552-1574. (https://doi.org/10.1137/S1064827595295337)

source
OrdinaryDiffEqExponentialRK.EPIRK4s3A — Type
EPIRK4s3A(; adaptive_krylov = true,
+            m = 30,
+            iop = 0)

Semilinear ODE solver 4th order EPIRK scheme with stiff order 4.

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Rainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)

source
OrdinaryDiffEqExponentialRK.EPIRK4s3B — Type
EPIRK4s3B(; adaptive_krylov = true,
+            m = 30,
+            iop = 0)

Semilinear ODE solver 4th order EPIRK scheme with stiff order 4.

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Rainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)

source
OrdinaryDiffEqExponentialRK.EPIRK5s3 — Type
EPIRK5s3(; adaptive_krylov = true,
+           m = 30,
+           iop = 0)

Semilinear ODE solver 5th order “horizontal” EPIRK scheme with stiff order 5. Broken.

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Rainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)

source
OrdinaryDiffEqExponentialRK.EXPRB53s3 — Type
EXPRB53s3(; adaptive_krylov = true,
+            m = 30,
+            iop = 0)

Semilinear ODE solver 5th order EPIRK scheme with stiff order 5.

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Rainwater, G., & Tokman, M. (2016). A new approach to constructing efficient stiffly accurate EPIRK methods. Journal of Computational Physics, 323, 283-309. (https://doi.org/10.1016/j.jcp.2016.07.026)

source
OrdinaryDiffEqExponentialRK.EPIRK5P1 — Type
EPIRK5P1(; adaptive_krylov = true,
+           m = 30,
+           iop = 0)

Semilinear ODE solver 5th order EPIRK scheme

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Tokman, M., Loffeld, J., & Tranquilli, P. (2012). New Adaptive Exponential Propagation Iterative Methods of Runge–Kutta Type. SIAM Journal on Scientific Computing, 34(5), A2650-A2669. (https://doi.org/10.1137/110849961)

source
OrdinaryDiffEqExponentialRK.EPIRK5P2 — Type
EPIRK5P2(; adaptive_krylov = true,
+           m = 30,
+           iop = 0)

Semilinear ODE solver 5th order EPIRK scheme

Keyword Arguments

  • adaptive_krylov: Determines if the adaptive Krylov algorithm with timestepping of Neisen & Wright is used.
  • m: Controls the size of Krylov subspace. - iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Tokman, M., Loffeld, J., & Tranquilli, P. (2012). New Adaptive Exponential Propagation Iterative Methods of Runge–Kutta Type. SIAM Journal on Scientific Computing, 34(5), A2650-A2669. (https://doi.org/10.1137/110849961)

source
diff --git a/dev/semilinear/Linear/index.html b/dev/semilinear/Linear/index.html new file mode 100644 index 0000000000..548c4d5d8d --- /dev/null +++ b/dev/semilinear/Linear/index.html @@ -0,0 +1,51 @@ + +OrdinaryDiffEqLinear · OrdinaryDiffEq.jl
GitHub

OrdinaryDiffEqLinear

Methods for semi-linear differential equations.

Installation

To be able to access the solvers in OrdinaryDiffEqLinear, you must first install them use the Julia package manager:

using Pkg
+Pkg.add("OrdinaryDiffEqLinear")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqLinear, SciMLOperators
+function update_func(A, u, p, t)
+    A[1, 1] = 0
+    A[2, 1] = sin(u[1])
+    A[1, 2] = -1
+    A[2, 2] = 0
+end
+A0 = ones(2, 2)
+A = DiffEqArrayOperator(A0, update_func = update_func)
+u0 = ones(2)
+tspan = (0.0, 30.0)
+prob = ODEProblem(A, u0, tspan)
+sol = solve(prob, LieRK4(), dt = 1 / 4)

Full list of solvers

Time and State-Independent Solvers

OrdinaryDiffEqLinear.LinearExponential — Type
LinearExponential(; krylov = :off,
+                    m = 10,
+                    iop = 0)

Semilinear ODE solver Exact solution formula for linear, time-independent problems.

Keyword Arguments

  • krylov:
    • :off: cache the operator beforehand. Requires Matrix(A) method defined for the operator A.
    • :simple: uses simple Krylov approximations with fixed subspace size m.
    • :adaptive: uses adaptive Krylov approximations with internal timestepping.
  • m: Controls the size of Krylov subspace if krylov=:simple, and the initial subspace size if krylov=:adaptive.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@book{strogatz2018nonlinear, title={Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering}, author={Strogatz, Steven H}, year={2018}, publisher={CRC press}}

source

Time-Dependent and State-Independent Solvers

OrdinaryDiffEqLinear.MagnusMidpoint — Type
MagnusMidpoint(; krylov = false,
+                 m = 30,
+                 iop = 0)

Semilinear ODE solver Second order Magnus Midpoint method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

https://joshuagoings.com/2017/06/15/magnus/

source
OrdinaryDiffEqLinear.MagnusLeapfrog — Type
MagnusLeapfrog(; krylov = false,
+                 m = 30,
+                 iop = 0)

Semilinear ODE solver Second order Magnus Leapfrog method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

https://joshuagoings.com/2017/06/15/magnus/

source
OrdinaryDiffEqLinear.MagnusGauss4 — Type
MagnusGauss4(; krylov = false,
+               m = 30,
+               iop = 0)

Semilinear ODE solver Fourth order Magnus method approximated using a two stage Gauss quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{hairer2011solving, title={Solving differential equations on manifolds}, author={Hairer, Ernst}, journal={Lecture notes}, year={2011} }

source
OrdinaryDiffEqLinear.MagnusNC6 — Type
MagnusNC6(; krylov = false,
+            m = 30,
+            iop = 0)

Semilinear ODE solver Sixth order Magnus method approximated using Newton-Cotes quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }

source
OrdinaryDiffEqLinear.MagnusGL6 — Type
MagnusGL6(; krylov = false,
+            m = 30,
+            iop = 0)

Semilinear ODE solver Sixth order Magnus method approximated using Gauss-Legendre quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }

source
OrdinaryDiffEqLinear.MagnusGL8 — Type
MagnusGL8(; krylov = false,
+            m = 30,
+            iop = 0)

Semilinear ODE solver Eighth order Magnus method approximated using Newton-Cotes quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }

source
OrdinaryDiffEqLinear.MagnusNC8 — Type
MagnusNC8(; krylov = false,
+            m = 30,
+            iop = 0)

Semilinear ODE solver Eighth order Magnus method approximated using Gauss-Legendre quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{blanes2000improved, title={Improved high order integrators based on the Magnus expansion}, author={Blanes, Sergio and Casas, Fernando and Ros, Javier}, journal={BIT Numerical Mathematics}, volume={40}, number={3}, pages={434–450}, year={2000}, publisher={Springer} }

source
OrdinaryDiffEqLinear.MagnusGL4 — Type
MagnusGL4(; krylov = false,
+            m = 30,
+            iop = 0)

Semilinear ODE solver Fourth order Magnus method approximated using Gauss-Legendre quadrature.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{blanes2009magnus, title={The Magnus expansion and some of its applications}, author={Blanes, Sergio and Casas, Fernando and Oteo, Jose-Angel and Ros, Jos{'e}}, journal={Physics reports}, volume={470}, number={5-6}, pages={151–238}, year={2009}, publisher={Elsevier} }

source

State-Dependent Solvers

OrdinaryDiffEqLinear.LieEuler — Type
LieEuler(; krylov = false,
+           m = 30,
+           iop = 0)

Semilinear ODE solver description

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }

source
OrdinaryDiffEqLinear.RKMK2 — Type
RKMK2(; krylov = false,
+        m = 30,
+        iop = 0)

Semilinear ODE solver Second order Runge–Kutta–Munthe-Kaas method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }

source
OrdinaryDiffEqLinear.RKMK4 — Type
RKMK4(; krylov = false,
+        m = 30,
+        iop = 0)

Semilinear ODE solver Fourth order Runge–Kutta–Munthe-Kaas method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }

source
OrdinaryDiffEqLinear.LieRK4 — Type
LieRK4(; krylov = false,
+         m = 30,
+         iop = 0)

Semilinear ODE solver Fourth order Lie Runge-Kutta method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }

source
OrdinaryDiffEqLinear.CG2 — Type
CG2(; krylov = false,
+      m = 30,
+      iop = 0)

Semilinear ODE solver Second order Crouch–Grossman method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{celledoni2014introduction, title={An introduction to Lie group integrators–basics, new developments and applications}, author={Celledoni, Elena and Marthinsen, H{a}kon and Owren, Brynjulf}, journal={Journal of Computational Physics}, volume={257}, pages={1040–1061}, year={2014}, publisher={Elsevier} }

source
OrdinaryDiffEqLinear.CG4a — Type
CG4a(; krylov = false,
+       m = 30,
+       iop = 0)

Semilinear ODE solver Fourth order Crouch-Grossman method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{jackiewicz2000construction, title={Construction of Runge–Kutta methods of Crouch–Grossman type of high order}, author={Jackiewicz, Zdzislaw and Marthinsen, Arne and Owren, Brynjulf}, journal={Advances in Computational Mathematics}, volume={13}, pages={405–415}, year={2000}, publisher={Springer} }

source
OrdinaryDiffEqLinear.MagnusAdapt4 — Type
MagnusAdapt4()

Semilinear ODE solver Fourth Order Adaptive Magnus method.

Keyword Arguments

References

@article{li2008adaptive, title={Adaptive explicit Magnus numerical method for nonlinear dynamical systems}, author={Li, Wen-cheng and Deng, Zi-chen}, journal={Applied Mathematics and Mechanics}, volume={29}, number={9}, pages={1111–1118}, year={2008}, publisher={Springer}}

source
OrdinaryDiffEqLinear.CayleyEuler — Type
CayleyEuler()

Semilinear ODE solver First order method using Cayley transformations.

Keyword Arguments

References

@article{iserles2000lie, title={Lie-group methods}, author={Iserles, Arieh and Munthe-Kaas, Hans Z and Norsett, Syvert P and Zanna, Antonella}, journal={Acta numerica}, volume={9}, pages={215–365}, year={2000}, publisher={Cambridge University Press}}

source

Time and State-Dependent Operators

OrdinaryDiffEqLinear.CG3 — Type
CG3(; krylov = false,
+      m = 30,
+      iop = 0)

Semilinear ODE solver Third order Crouch-Grossman method.

Keyword Arguments

  • krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
  • m: Controls the size of Krylov subspace.
  • iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@article{crouch1993numerical, title={Numerical integration of ordinary differential equations on manifolds}, author={Crouch, Peter E and Grossman, R}, journal={Journal of Nonlinear Science}, volume={3}, pages={1–33}, year={1993}, publisher={Springer} }

source
diff --git a/dev/semilinear/exponential_rk/index.html b/dev/semilinear/exponential_rk/index.html deleted file mode 100644 index 51245370a0..0000000000 --- a/dev/semilinear/exponential_rk/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Exponential Runge-Kutta Integrators · OrdinaryDiffEq.jl
GitHub

Exponential Runge-Kutta Integrators

Missing docstring.

Missing docstring for LawsonEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for NorsettEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ETD2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ETDRK2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ETDRK3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ETDRK4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for HochOst4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Exp4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK4s3A. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK4s3B. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5s3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EXPRB53s3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5P1. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5P2. Check Documenter's build log for details.

diff --git a/dev/semilinear/magnus/index.html b/dev/semilinear/magnus/index.html deleted file mode 100644 index ffe1c25770..0000000000 --- a/dev/semilinear/magnus/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Magnus and Lie Group Integrators · OrdinaryDiffEq.jl
GitHub

Magnus and Lie Group Integrators

Missing docstring.

Missing docstring for MagnusMidpoint. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusLeapfrog. Check Documenter's build log for details.

Missing docstring.

Missing docstring for LieEuler. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusGauss4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusNC6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusGL6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusGL8. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusNC8. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusGL4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RKMK2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RKMK4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for LieRK4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CG2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CG3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CG4a. Check Documenter's build log for details.

Missing docstring.

Missing docstring for MagnusAdapt4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for CayleyEuler. Check Documenter's build log for details.

diff --git a/dev/stiff/firk/index.html b/dev/stiff/firk/index.html deleted file mode 100644 index 628e120078..0000000000 --- a/dev/stiff/firk/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Fully Implicit Runge-Kutta (FIRK) Methods · OrdinaryDiffEq.jl
GitHub

Fully Implicit Runge-Kutta (FIRK) Methods

OrdinaryDiffEqFIRK.RadauIIA3 — Type

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier} }

RadauIIA3: Fully-Implicit Runge-Kutta Method An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.

source
OrdinaryDiffEqFIRK.RadauIIA5 — Type

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier} }

RadauIIA5: Fully-Implicit Runge-Kutta Method An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.

source
diff --git a/dev/stiff/implicit_extrapolation/index.html b/dev/stiff/implicit_extrapolation/index.html deleted file mode 100644 index d737d917b1..0000000000 --- a/dev/stiff/implicit_extrapolation/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Implicit Extrapolation Methods · OrdinaryDiffEq.jl
GitHub

Implicit Extrapolation Methods

diff --git a/dev/stiff/rosenbrock/index.html b/dev/stiff/rosenbrock/index.html deleted file mode 100644 index ece7165757..0000000000 --- a/dev/stiff/rosenbrock/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Rosenbrock Methods · OrdinaryDiffEq.jl
GitHub

Rosenbrock Methods

Standard Rosenbrock Methods

Missing docstring.

Missing docstring for ROS2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS2PR. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS3PR. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Scholz4_7. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS3PRL. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS3PRL2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS3P. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas3P. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RosShamp4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Veldd4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Velds4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for GRK4T. Check Documenter's build log for details.

Missing docstring.

Missing docstring for GRK4A. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Ros4LStab. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas42. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas4P. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas4P2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas5P. Check Documenter's build log for details.

Rosenbrock W-Methods

Missing docstring.

Missing docstring for Rosenbrock23. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rosenbrock32. Check Documenter's build log for details.

Missing docstring.

Missing docstring for Rodas23W. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS34PW1a. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS34PW1b. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS34PW2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS34PW3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS34PRw. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROK4a. Check Documenter's build log for details.

Missing docstring.

Missing docstring for ROS2S. Check Documenter's build log for details.

Missing docstring.

Missing docstring for RosenbrockW6S4OS. Check Documenter's build log for details.

diff --git a/dev/stiff/sdirk/index.html b/dev/stiff/sdirk/index.html deleted file mode 100644 index fc65643663..0000000000 --- a/dev/stiff/sdirk/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods · OrdinaryDiffEq.jl
GitHub

Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods

OrdinaryDiffEqSDIRK.ImplicitEuler — Type

ImplicitEuler: SDIRK Method A 1st order implicit solver. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).

source
OrdinaryDiffEqSDIRK.ImplicitMidpoint — Type

ImplicitMidpoint: SDIRK Method A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.

source
OrdinaryDiffEqSDIRK.Trapezoid — Type

Andre Vladimirescu. 1994. The Spice Book. John Wiley & Sons, Inc., New York, NY, USA.

Trapezoid: SDIRK Method A second order A-stable symmetric ESDIRK method. "Almost symplectic" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences approximation to the second derivative terms in the local truncation error estimate (the SPICE approximation strategy).

source
OrdinaryDiffEqSDIRK.TRBDF2 — Type

@article{hosea1996analysis, title={Analysis and implementation of TR-BDF2}, author={Hosea, ME and Shampine, LF}, journal={Applied Numerical Mathematics}, volume={20}, number={1-2}, pages={21–37}, year={1996}, publisher={Elsevier} }

TRBDF2: SDIRK Method A second order A-B-L-S-stable one-step ESDIRK method. Includes stiffness-robust error estimates for accurate adaptive timestepping, smoothed derivatives for highly stiff and oscillatory problems.

source
OrdinaryDiffEqSDIRK.SDIRK2 — Type

@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM} }

SDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method

source
Missing docstring.

Missing docstring for SDIRK22. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SSPSDIRK2. Check Documenter's build log for details.

OrdinaryDiffEqSDIRK.Kvaerno3 — Type

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }

Kvaerno3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method

source
Missing docstring.

Missing docstring for CFNLIRK3. Check Documenter's build log for details.

OrdinaryDiffEqSDIRK.Cash4 — Type

@article{hindmarsh2005sundials, title={{SUNDIALS}: Suite of nonlinear and differential/algebraic equation solvers}, author={Hindmarsh, Alan C and Brown, Peter N and Grant, Keith E and Lee, Steven L and Serban, Radu and Shumaker, Dan E and Woodward, Carol S}, journal={ACM Transactions on Mathematical Software (TOMS)}, volume={31}, number={3}, pages={363–396}, year={2005}, publisher={ACM} }

Cash4: SDIRK Method An A-L stable 4th order SDIRK method

source
Missing docstring.

Missing docstring for SFSDIRK4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SFSDIRK5. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SFSDIRK6. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SFSDIRK7. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SFSDIRK8. Check Documenter's build log for details.

OrdinaryDiffEqSDIRK.Hairer4 — Type

E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

Hairer4: SDIRK Method An A-L stable 4th order SDIRK method

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OrdinaryDiffEqSDIRK.Hairer42 — Type

E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

Hairer42: SDIRK Method An A-L stable 4th order SDIRK method

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OrdinaryDiffEqSDIRK.Kvaerno4 — Type

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }

Kvaerno4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method.

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OrdinaryDiffEqSDIRK.Kvaerno5 — Type

@article{kvaerno2004singly, title={Singly diagonally implicit Runge–Kutta methods with an explicit first stage}, author={Kv{\ae}rn{\o}, Anne}, journal={BIT Numerical Mathematics}, volume={44}, number={3}, pages={489–502}, year={2004}, publisher={Springer} }

Kvaerno5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method

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diff --git a/dev/stiff/stabilized_rk/index.html b/dev/stiff/stabilized_rk/index.html deleted file mode 100644 index d77b20559a..0000000000 --- a/dev/stiff/stabilized_rk/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev) · OrdinaryDiffEq.jl
GitHub

Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)

Explicit Stabilized Runge-Kutta Methods

Explicit stabilized methods utilize an upper bound on the spectral radius of the Jacobian. Users can supply an upper bound by specifying the keyword argument eigen_est, for example

`eigen_est = (integrator) -> integrator.eigen_est = upper_bound`

The methods ROCK2 and ROCK4 also include keyword arguments min_stages and max_stages, which specify upper and lower bounds on the adaptively chosen number of stages for stability.

OrdinaryDiffEqStabilizedRK.ROCK2 — Type

Assyr Abdulle, Alexei A. Medovikov. Second Order Chebyshev Methods based on Orthogonal Polynomials. Numerische Mathematik, 90 (1), pp 1-18, 2001. doi: https://dx.doi.org/10.1007/s002110100292

ROCK2: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

source
OrdinaryDiffEqStabilizedRK.ROCK4 — Type
ROCK4(; min_stages = 0, max_stages = 152, eigen_est = nothing)

Assyr Abdulle. Fourth Order Chebyshev Methods With Recurrence Relation. 2002 Society for Industrial and Applied Mathematics Journal on Scientific Computing, 23(6), pp 2041-2054, 2001. doi: https://doi.org/10.1137/S1064827500379549

ROCK4: Stabilized Explicit Method. Fourth order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes optional keyword arguments min_stages, max_stages, and eigen_est. The function eigen_est should be of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

source
Missing docstring.

Missing docstring for SERK2. Check Documenter's build log for details.

OrdinaryDiffEqStabilizedRK.ESERK4 — Type
ESERK4(; eigen_est = nothing)

J. Martín-Vaquero, B. Kleefeld. Extrapolated stabilized explicit Runge-Kutta methods, Journal of Computational Physics, 326, pp 141-155, 2016. doi: https://doi.org/10.1016/j.jcp.2016.08.042.

ESERK4: Stabilized Explicit Method. Fourth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

source
OrdinaryDiffEqStabilizedRK.ESERK5 — Type
ESERK5(; eigen_est = nothing)

J. Martín-Vaquero, A. Kleefeld. ESERK5: A fifth-order extrapolated stabilized explicit Runge-Kutta method, Journal of Computational and Applied Mathematics, 356, pp 22-36, 2019. doi: https://doi.org/10.1016/j.cam.2019.01.040.

ESERK5: Stabilized Explicit Method. Fifth order extrapolated stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues and is smoothened to allow for moderate sized complex eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

source
OrdinaryDiffEqStabilizedRK.RKC — Type
RKC(; eigen_est = nothing)

B. P. Sommeijer, L. F. Shampine, J. G. Verwer. RKC: An Explicit Solver for Parabolic PDEs, Journal of Computational and Applied Mathematics, 88(2), pp 315-326, 1998. doi: https://doi.org/10.1016/S0377-0427(97)00219-7

RKC: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues.

This method takes the keyword argument eigen_est of the form

eigen_est = (integrator) -> integrator.eigen_est = upper_bound,

where upper_bound is an estimated upper bound on the spectral radius of the Jacobian matrix. If eigen_est is not provided, upper_bound will be estimated using the power iteration.

source

Implicit Stabilized Runge-Kutta Methods

Missing docstring.

Missing docstring for IRKC. Check Documenter's build log for details.

diff --git a/dev/stiff/stiff_multistep/index.html b/dev/stiff/stiff_multistep/index.html deleted file mode 100644 index 188d8354b5..0000000000 --- a/dev/stiff/stiff_multistep/index.html +++ /dev/null @@ -1,6 +0,0 @@ - -Multistep Methods for Stiff Equations · OrdinaryDiffEq.jl
GitHub

Multistep Methods for Stiff Equations

OrdinaryDiffEqBDF.QNDF1 — Type

QNDF1: Multistep Method An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine's accuracy-optimal -0.1850.

See also QNDF.

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OrdinaryDiffEqBDF.QNDF2 — Type

QNDF2: Multistep Method An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.

See also QNDF.

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OrdinaryDiffEqBDF.ABDF2 — Type

E. Alberdi Celayaa, J. J. Anza Aguirrezabalab, P. Chatzipantelidisc. Implementation of an Adaptive BDF2 Formula and Comparison with The MATLAB Ode15s. Procedia Computer Science, 29, pp 1014-1026, 2014. doi: https://doi.org/10.1016/j.procs.2014.05.091

ABDF2: Multistep Method An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.

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OrdinaryDiffEqBDF.QNDF — Type

QNDF: Multistep Method An adaptive order quasi-constant timestep NDF method. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).

@article{shampine1997matlab, title={The matlab ode suite}, author={Shampine, Lawrence F and Reichelt, Mark W}, journal={SIAM journal on scientific computing}, volume={18}, number={1}, pages={1–22}, year={1997}, publisher={SIAM} }

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OrdinaryDiffEqBDF.FBDF — Type

FBDF: Fixed leading coefficient BDF

An adaptive order quasi-constant timestep NDF method. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).

@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH \& Co. KG} }

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OrdinaryDiffEqBDF.MEBDF2 — Type

MEBDF2: Multistep Method The second order Modified Extended BDF method, which has improved stability properties over the standard BDF. Fixed timestep only.

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diff --git a/dev/usage/index.html b/dev/usage/index.html index c54515c9a5..928487213d 100644 --- a/dev/usage/index.html +++ b/dev/usage/index.html @@ -3,7 +3,7 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Usage

OrdinaryDiffEq.jl is part of the SciML common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an OrdinaryDiffEq.jl algorithm to solve. For example, we can solve the ODE tutorial from the docs using the Tsit5() algorithm:

using OrdinaryDiffEq
+
GitHub
Usage
OrdinaryDiffEq.jl is part of the SciML common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an OrdinaryDiffEq.jl algorithm to solve. For example, we can solve the ODE tutorial from the docs using the Tsit5() algorithm:
using OrdinaryDiffEq
 f(u, p, t) = 1.01 * u
 u0 = 1 / 2
 tspan = (0.0, 1.0)
@@ -39,4 +39,4 @@
 initial_positions = [0.0, 0.1]
 initial_velocities = [0.5, 0.0]
 prob = SecondOrderODEProblem(HH_acceleration, initial_velocities, initial_positions, tspan)
-sol2 = solve(prob, KahanLi8(), dt = 1 / 10);
Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).
Available Solvers
For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.

+sol2 = solve(prob, KahanLi8(), dt = 1 / 10);

Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).

Available Solvers

For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.