diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index d522dbcb..d5ddb1b1 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-04T23:18:12","documenter_version":"1.7.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-04T23:22:28","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/ConservationLaws/index.html b/dev/ConservationLaws/index.html index 40be18cb..6401d795 100644 --- a/dev/ConservationLaws/index.html +++ b/dev/ConservationLaws/index.html @@ -2,7 +2,7 @@ ConservationLaws · StableSpectralElements.jl
GitHub

Module ConservationLaws

Overview

The equations to be solved are defined by subtypes of AbstractConservationLaw on which functions such as physical_flux and numerical_flux are dispatched. Objects of type AbstractConservationLaw contain two type parameters, d and PDEType, the former denoting the spatial dimension of the problem, which is inherited by all subtypes, and the latter being a subtype of AbstractPDEType denoting the particular type of PDE being solved, which is either FirstOrder or SecondOrder. Whereas first-order problems remove the dependence of the flux tensor on the solution gradient in order to obtain systems of the form

\[\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t)) = \underline{0},\]

second-order problems are treated by StableSpectralElements.jl as first-order systems of the form

\[\begin{aligned} \underline{\bm{Q}}(\bm{x},t) - \bm{\nabla}_{\bm{x}} \underline{U}(\bm{x},t) &= \underline{0},\\ \partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \underline{\bm{Q}}(\bm{x},t)) &= \underline{0}. -\end{aligned}\]

Currently, the linear advection and advection-diffusion equations, the inviscid and viscous Burgers' equations, and the compressible Euler equations are supported by StableSpectralElements.jl, but any system of the above form can in principle be implemented, provided that appropriate physical and numerical fluxes are defined.

Equations

Listed below are partial differential equations supported by StableSpectralElements.jl.

StableSpectralElements.ConservationLaws.LinearAdvectionEquation — Type
LinearAdvectionEquation(a::NTuple{d,Float64}) where {d}

Define a linear advection equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) \big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$. A specialized constructor LinearAdvectionEquation(a::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation — Type
LinearAdvectionDiffusionEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a linear advection-diffusion equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$ and diffusion coefficient $b \in \R^+$. A specialized constructor LinearAdvectionDiffusionEquation(a::Float64, b::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.InviscidBurgersEquation — Type
InviscidBurgersEquation(a::NTuple{d,Float64}) where {d}

Define an inviscid Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 \big) = 0,\]

where $\bm{a} \in \R^d$. A specialized constructor InviscidBurgersEquation() is provided for the one-dimensional case with a = (1.0,).

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StableSpectralElements.ConservationLaws.ViscousBurgersEquation — Type

x ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a viscous Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

where $\bm{a} \in \R^d$ and $b \in \R^+$. A specialized constructor ViscousBurgersEquation(b::Float64) is provided for the one-dimensional case with a = (1.0,).

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StableSpectralElements.ConservationLaws.EulerEquations — Type
EulerEquations{d}(γ::Float64) where {d}

Define an Euler system governing compressible, adiabatic fluid flow, taking the form

\[\frac{\partial}{\partial t}\left[\begin{array}{c} +\end{aligned}\]

Currently, the linear advection and advection-diffusion equations, the inviscid and viscous Burgers' equations, and the compressible Euler equations are supported by StableSpectralElements.jl, but any system of the above form can in principle be implemented, provided that appropriate physical and numerical fluxes are defined.

Equations

Listed below are partial differential equations supported by StableSpectralElements.jl.

StableSpectralElements.ConservationLaws.LinearAdvectionEquation — Type
LinearAdvectionEquation(a::NTuple{d,Float64}) where {d}

Define a linear advection equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) \big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$. A specialized constructor LinearAdvectionEquation(a::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation — Type
LinearAdvectionDiffusionEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a linear advection-diffusion equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$ and diffusion coefficient $b \in \R^+$. A specialized constructor LinearAdvectionDiffusionEquation(a::Float64, b::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.InviscidBurgersEquation — Type
InviscidBurgersEquation(a::NTuple{d,Float64}) where {d}

Define an inviscid Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 \big) = 0,\]

where $\bm{a} \in \R^d$. A specialized constructor InviscidBurgersEquation() is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.ViscousBurgersEquation — Type

x ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a viscous Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

where $\bm{a} \in \R^d$ and $b \in \R^+$. A specialized constructor ViscousBurgersEquation(b::Float64) is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.EulerEquations — Type
EulerEquations{d}(γ::Float64) where {d}

Define an Euler system governing compressible, adiabatic fluid flow, taking the form

\[\frac{\partial}{\partial t}\left[\begin{array}{c} \rho(\bm{x}, t) \\ \rho(\bm{x}, t) V_1(\bm{x}, t) \\ \vdots \\ @@ -14,4 +14,4 @@ \vdots \\ \rho(\bm{x}, t) V_d(\bm{x}, t) V_m(\bm{x}, t)+P(\bm{x}, t) \delta_{d m} \\ V_m(\bm{x}, t)(E(\bm{x}, t)+P(\bm{x}, t)) -\end{array}\right]=\underline{0},\]

where $\rho(\bm{x},t) \in \mathbb{R}$ is the fluid density, $\bm{V}(\bm{x},t) \in \mathbb{R}^d$ is the flow velocity, $E(\bm{x},t) \in \mathbb{R}$ is the total energy per unit volume, and the pressure is given for an ideal gas with constant specific heat as

\[P(\bm{x},t) = (\gamma - 1)\Big(E(\bm{x},t) - \frac{1}{2}\rho(\bm{x},t) \lVert \bm{V}(\bm{x},t)\rVert^2\Big).\]

The specific heat ratio is specified as a parameter γ::Float64, which must be greater than unity.

source
+\end{array}\right]=\underline{0},\]

where $\rho(\bm{x},t) \in \mathbb{R}$ is the fluid density, $\bm{V}(\bm{x},t) \in \mathbb{R}^d$ is the flow velocity, $E(\bm{x},t) \in \mathbb{R}$ is the total energy per unit volume, and the pressure is given for an ideal gas with constant specific heat as

\[P(\bm{x},t) = (\gamma - 1)\Big(E(\bm{x},t) - \frac{1}{2}\rho(\bm{x},t) \lVert \bm{V}(\bm{x},t)\rVert^2\Big).\]

The specific heat ratio is specified as a parameter γ::Float64, which must be greater than unity.

source diff --git a/dev/SpatialDiscretizations/index.html b/dev/SpatialDiscretizations/index.html index 9913404c..5972c0c1 100644 --- a/dev/SpatialDiscretizations/index.html +++ b/dev/SpatialDiscretizations/index.html @@ -6,4 +6,4 @@ \hat{\Omega}_{\mathrm{tri}} &= \big\{ \bm{\xi} \in [-1,1]^2 : \xi_1 + \xi_2 \leq 0 \big\},\\ \hat{\Omega}_{\mathrm{tet}} &= \big\{ \bm{\xi} \in [-1,1]^3 : \xi_1 + \xi_2 + \xi_3 \leq -1 \big\}. \end{aligned}\]

These element types are used in the constructor for StableSpectralElements.jl's ReferenceApproximation type, along with a subtype of AbstractApproximationType specifying the nature of the local approximation (and, optionally, the associated volume and facet quadrature rules).

All the information used to define the spatial discretization on the physical domain $\Omega$ is contained within a SpatialDiscretization structure, which is constructed using a ReferenceApproximation and a MeshData from StartUpDG.jl, which are stored as the fields reference_approximation and mesh. When the constructor for a SpatialDiscretization is called, the grid metrics are computed and stored in a GeometricFactors structure, with the corresponding field being geometric_factors.

Data types

StableSpectralElements.SpatialDiscretizations.ReferenceApproximation — Type
ReferenceApproximation(approx_type, reference_element, D, V, Vf, R, W, B, V_plot,
-                       reference_mapping)

Data structure defining the discretization on the reference element, containing the following fields:

  • approx_type::AbstractApproximationType: Type of operators used for the discretization on the reference element(NodalTensor, ModalTensor, NodalMulti, ModalMulti NodalMultiDiagE, ModalMultiDiagE)
  • reference_element::StartUpDG.RefElemData: Data structure containing quadrature node positions and operators used for defining the mapping from reference to physical space; contains the field element_type::StartUpDG.AbstractElemShape which determines the shape of the reference element (currently, StableSpectralElements.jl supports the options Line,Tri, Quad, Hex, and Tet)
  • D::NTuple{d, <:LinearMap}: Tuple of operators of size N_q by N_q approximating each partial derivative at the volume quadrature nodes
  • V::LinearMap: Generalized Vandermonde matrix of size N_q by N_p mapping solution degrees of freedom to values at volume quadrature nodes
  • Vf::LinearMap: Generalized Vandermonde matrix of size N_f by N_p mapping solution degrees of freedom to values at facet quadrature nodes
  • R::LinearMap: Interpolation/extrapolation operator of size N_f by N_q which maps nodal data from volume quadrature nodes to facet quadrature nodes
  • W::Diagonal: Volume quadrature weight matrix of size N_q by N_q
  • B::Diagonal: Facet quadrature weight matrix of size N_f by N_f
  • V_plot::LinearMap: Generalized Vandermonde matrix mapping solution degrees of freedom to plotting nodes
  • reference_mapping::AbstractReferenceMapping: Optional collapsed coordinate transformation (either ReferenceMapping or NoMapping); if such a mapping is used (i.e. not NoMapping), the discrete derivative operators approximate partial derivatives with respect to components of the collapsed coordinate system

Outer constructors are provided to construct the discrete operators by dispatching on each combination of subtypes of AbstractApproximationTypeandAbstractElementShape`.

source

Approximation types

Listed below are approximation types (i.e. subtypes of AbstractApproximationType) supported by StableSpectralElements.jl.

Tensor-product operators

The following subtypes of AbstractTensorProduct <: AbstractApproximationType employ tensor-product operators.

StableSpectralElements.SpatialDiscretizations.NodalTensor — Type
NodalTensor(p::Int)

Approximation type for a nodal formulation of polynomial degree $p$ based on tensor-product volume and facet quadrature rules (generalized Vandermonde matrix is identity, derivative and interpolation/extrapolation operators have tensor-product structure).

source
StableSpectralElements.SpatialDiscretizations.ModalTensor — Type
ModalTensor(p::Int)

Approximation type for a modal formulation of polynomial degree $p$ based on tensor-product volume and facet quadrature rules (generalized Vandermonde matrix is not necessarily identity, derivative and interpolation/extrapolation operators have tensor-product structure).

source

Multidimensional operators

The following subtypes of AbstractMultidimensional <: AbstractApproximationType employ multidimensional operators.

StableSpectralElements.SpatialDiscretizations.NodalMulti — Type
NodalMulti(p::Int)

Approximation type for a nodal formulation based on multidimensional volume and facet quadrature rules (generalized Vandermonde matrix is identity, derivative and interpolation/extrapolation operators are dense).

source
StableSpectralElements.SpatialDiscretizations.ModalMulti — Type
ModalMulti(p::Int)

Approximation type for a modal formulation of polynomial degree $p$ based on multidimensional volume and facet quadrature rules (generalized Vandermonde, derivative and interpolation/extrapolation operators are all dense).

source
StableSpectralElements.SpatialDiscretizations.NodalMultiDiagE — Type
NodalMultiDiagE(p::Int)

Approximation type for a nodal formulation of polynomial degree $p$ based on a multidimensional volume quadrature rule including nodes collocated with those used for facet integration (generalized Vandermonde matrix is identity, derivative operator is dense, interpolation/extrapolation operator picks out values at facet quadrature nodes).

source
StableSpectralElements.SpatialDiscretizations.ModalMultiDiagE — Type
ModalMultiDiagE(p::Int)

Approximation type for a modal formulation based on a multidimensional volume quadrature rule of polynomial degree $p$ including nodes collocated with those used for facet integration (generalized Vandermonde and derivative operators are dense, interpolation/extrapolation operator picks out values at facet quadrature nodes).

source
+ reference_mapping)

Data structure defining the discretization on the reference element, containing the following fields:

Outer constructors are provided to construct the discrete operators by dispatching on each combination of subtypes of AbstractApproximationTypeandAbstractElementShape`.

source

Approximation types

Listed below are approximation types (i.e. subtypes of AbstractApproximationType) supported by StableSpectralElements.jl.

Tensor-product operators

The following subtypes of AbstractTensorProduct <: AbstractApproximationType employ tensor-product operators.

StableSpectralElements.SpatialDiscretizations.NodalTensor — Type
NodalTensor(p::Int)

Approximation type for a nodal formulation of polynomial degree $p$ based on tensor-product volume and facet quadrature rules (generalized Vandermonde matrix is identity, derivative and interpolation/extrapolation operators have tensor-product structure).

source
StableSpectralElements.SpatialDiscretizations.ModalTensor — Type
ModalTensor(p::Int)

Approximation type for a modal formulation of polynomial degree $p$ based on tensor-product volume and facet quadrature rules (generalized Vandermonde matrix is not necessarily identity, derivative and interpolation/extrapolation operators have tensor-product structure).

source

Multidimensional operators

The following subtypes of AbstractMultidimensional <: AbstractApproximationType employ multidimensional operators.

StableSpectralElements.SpatialDiscretizations.NodalMulti — Type
NodalMulti(p::Int)

Approximation type for a nodal formulation based on multidimensional volume and facet quadrature rules (generalized Vandermonde matrix is identity, derivative and interpolation/extrapolation operators are dense).

source
StableSpectralElements.SpatialDiscretizations.ModalMulti — Type
ModalMulti(p::Int)

Approximation type for a modal formulation of polynomial degree $p$ based on multidimensional volume and facet quadrature rules (generalized Vandermonde, derivative and interpolation/extrapolation operators are all dense).

source
StableSpectralElements.SpatialDiscretizations.NodalMultiDiagE — Type
NodalMultiDiagE(p::Int)

Approximation type for a nodal formulation of polynomial degree $p$ based on a multidimensional volume quadrature rule including nodes collocated with those used for facet integration (generalized Vandermonde matrix is identity, derivative operator is dense, interpolation/extrapolation operator picks out values at facet quadrature nodes).

source
StableSpectralElements.SpatialDiscretizations.ModalMultiDiagE — Type
ModalMultiDiagE(p::Int)

Approximation type for a modal formulation based on a multidimensional volume quadrature rule of polynomial degree $p$ including nodes collocated with those used for facet integration (generalized Vandermonde and derivative operators are dense, interpolation/extrapolation operator picks out values at facet quadrature nodes).

source
diff --git a/dev/index.html b/dev/index.html index 7fd86f0e..9f1b176c 100644 --- a/dev/index.html +++ b/dev/index.html @@ -22,4 +22,4 @@ pages = {113360}, doi = {10.1016/j.jcp.2024.113360}, year = {2024} -}

License

This software is released under the GPLv3 license.

+}

License

This software is released under the GPLv3 license.

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