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diff --git a/previews/PR2390/dynamical/nystrom/index.html b/previews/PR2390/dynamical/nystrom/index.html
index 3623cd1663..05e61feda7 100644
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repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/dynamical/nystrom.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Runge-Kutta-Nystrom-Methods"><a class="docs-heading-anchor" href="#Runge-Kutta-Nystrom-Methods">Runge-Kutta Nystrom Methods</a><a id="Runge-Kutta-Nystrom-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Runge-Kutta-Nystrom-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.IRKN3" href="#OrdinaryDiffEqRKN.IRKN3"><code>OrdinaryDiffEqRKN.IRKN3</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IRKN3</code></pre><p>Improved Runge-Kutta-Nyström method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only.</p><p>Second order ODE should not depend on the first derivative.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L1-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.IRKN4" href="#OrdinaryDiffEqRKN.IRKN4"><code>OrdinaryDiffEqRKN.IRKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IRKN4</code></pre><p>Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only.</p><p>Second order ODE should not be dependent on the first derivative.</p><p>Recommended for smooth problems with expensive functions to evaluate.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L97-L113">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom4" href="#OrdinaryDiffEqRKN.Nystrom4"><code>OrdinaryDiffEqRKN.Nystrom4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom4</code></pre><p>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.</p><p>In case the ODE Problem is not dependent on the first derivative consider using <a href="#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent"><code>Nystrom4VelocityIndependent</code></a> to increase performance.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L18-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom4VelocityIndependent" href="#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent"><code>OrdinaryDiffEqRKN.Nystrom4VelocityIndependent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom4VelocityIdependent</code></pre><p>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).</p><p>More efficient then <a href="#OrdinaryDiffEqRKN.Nystrom4"><code>Nystrom4</code></a> on velocity independent problems, since less evaluations are needed.</p><p>Fixed time steps only.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L80-L94">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom5VelocityIndependent" href="#OrdinaryDiffEqRKN.Nystrom5VelocityIndependent"><code>OrdinaryDiffEqRKN.Nystrom5VelocityIndependent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom5VelocityIndependent</code></pre><p>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.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L116-L127">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.FineRKN4" href="#OrdinaryDiffEqRKN.FineRKN4"><code>OrdinaryDiffEqRKN.FineRKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FineRKN4()</code></pre><p>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.</p><p><strong>References</strong></p><pre><code class="nohighlight hljs">@article{fine1987low,
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repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/dynamical/nystrom.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Runge-Kutta-Nystrom-Methods"><a class="docs-heading-anchor" href="#Runge-Kutta-Nystrom-Methods">Runge-Kutta Nystrom Methods</a><a id="Runge-Kutta-Nystrom-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Runge-Kutta-Nystrom-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.IRKN3" href="#OrdinaryDiffEqRKN.IRKN3"><code>OrdinaryDiffEqRKN.IRKN3</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IRKN3</code></pre><p>Improved Runge-Kutta-Nyström method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only.</p><p>Second order ODE should not depend on the first derivative.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L1-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.IRKN4" href="#OrdinaryDiffEqRKN.IRKN4"><code>OrdinaryDiffEqRKN.IRKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IRKN4</code></pre><p>Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only.</p><p>Second order ODE should not be dependent on the first derivative.</p><p>Recommended for smooth problems with expensive functions to evaluate.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L97-L113">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom4" href="#OrdinaryDiffEqRKN.Nystrom4"><code>OrdinaryDiffEqRKN.Nystrom4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom4</code></pre><p>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.</p><p>In case the ODE Problem is not dependent on the first derivative consider using <a href="#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent"><code>Nystrom4VelocityIndependent</code></a> to increase performance.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L18-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom4VelocityIndependent" href="#OrdinaryDiffEqRKN.Nystrom4VelocityIndependent"><code>OrdinaryDiffEqRKN.Nystrom4VelocityIndependent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom4VelocityIdependent</code></pre><p>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).</p><p>More efficient then <a href="#OrdinaryDiffEqRKN.Nystrom4"><code>Nystrom4</code></a> on velocity independent problems, since less evaluations are needed.</p><p>Fixed time steps only.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L80-L94">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.Nystrom5VelocityIndependent" href="#OrdinaryDiffEqRKN.Nystrom5VelocityIndependent"><code>OrdinaryDiffEqRKN.Nystrom5VelocityIndependent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nystrom5VelocityIndependent</code></pre><p>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.</p><p><strong>References</strong></p><p>E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L116-L127">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.FineRKN4" href="#OrdinaryDiffEqRKN.FineRKN4"><code>OrdinaryDiffEqRKN.FineRKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FineRKN4()</code></pre><p>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.</p><p><strong>References</strong></p><pre><code class="nohighlight hljs">@article{fine1987low,
   title={Low order practical {R}unge-{K}utta-{N}ystr{&quot;o}m methods},
   author={Fine, Jerry Michael},
   journal={Computing},
@@ -12,7 +12,7 @@
   pages={281--297},
   year={1987},
   publisher={Springer}
-}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L34-L54">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.FineRKN5" href="#OrdinaryDiffEqRKN.FineRKN5"><code>OrdinaryDiffEqRKN.FineRKN5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FineRKN5()</code></pre><p>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.</p><p><strong>References</strong></p><pre><code class="nohighlight hljs">@article{fine1987low,
+}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L34-L54">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.FineRKN5" href="#OrdinaryDiffEqRKN.FineRKN5"><code>OrdinaryDiffEqRKN.FineRKN5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FineRKN5()</code></pre><p>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.</p><p><strong>References</strong></p><pre><code class="nohighlight hljs">@article{fine1987low,
   title={Low order practical {R}unge-{K}utta-{N}ystr{&quot;o}m methods},
   author={Fine, Jerry Michael},
   journal={Computing},
@@ -21,4 +21,4 @@
   pages={281--297},
   year={1987},
   publisher={Springer}
-}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L57-L77">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN6" href="#OrdinaryDiffEqRKN.DPRKN6"><code>OrdinaryDiffEqRKN.DPRKN6</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN6</code></pre><p>6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative. Free 6th order interpolant.</p><p><strong>References</strong></p><p>@article{dormand1987runge, title={Runge-kutta-nystrom triples}, author={Dormand, JR and Prince, PJ}, journal={Computers \&amp; Mathematics with Applications}, volume={13}, number={12}, pages={937–949}, year={1987}, publisher={Elsevier} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L167-L184">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN6FM" href="#OrdinaryDiffEqRKN.DPRKN6FM"><code>OrdinaryDiffEqRKN.DPRKN6FM</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN6FM</code></pre><p>6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Compared to <a href="#OrdinaryDiffEqRKN.DPRKN6"><code>DPRKN6</code></a>, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L187-L205">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN8" href="#OrdinaryDiffEqRKN.DPRKN8"><code>OrdinaryDiffEqRKN.DPRKN8</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN8</code></pre><p>8th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Not as efficient as <a href="#OrdinaryDiffEqRKN.DPRKN12"><code>DPRKN12</code></a> when high accuracy is needed, however this solver is competitive with <a href="#OrdinaryDiffEqRKN.DPRKN6"><code>DPRKN6</code></a> at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L208-L228">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN12" href="#OrdinaryDiffEqRKN.DPRKN12"><code>OrdinaryDiffEqRKN.DPRKN12</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN12</code></pre><p>12th order explicit Rugne-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Most efficient when high accuracy is needed.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L231-L250">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN4" href="#OrdinaryDiffEqRKN.ERKN4"><code>OrdinaryDiffEqRKN.ERKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN4</code></pre><p>Embedded 4(3) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic problems.</p><p><strong>References</strong></p><p>@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{&quot;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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L253-L273">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN5" href="#OrdinaryDiffEqRKN.ERKN5"><code>OrdinaryDiffEqRKN.ERKN5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN5</code></pre><p>Embedded 5(4) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic problems.</p><p><strong>References</strong></p><p>@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{&quot;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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L276-L297">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN7" href="#OrdinaryDiffEqRKN.ERKN7"><code>OrdinaryDiffEqRKN.ERKN7</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN7</code></pre><p>Embedded pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic Problems.</p><p><strong>References</strong></p><p>@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{&quot;o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L300-L318">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../stiff/implicit_extrapolation/">« Implicit Extrapolation Methods</a><a class="docs-footer-nextpage" href="../symplectic/">Symplectic Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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+}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L57-L77">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN6" href="#OrdinaryDiffEqRKN.DPRKN6"><code>OrdinaryDiffEqRKN.DPRKN6</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN6</code></pre><p>6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative. Free 6th order interpolant.</p><p><strong>References</strong></p><p>@article{dormand1987runge, title={Runge-kutta-nystrom triples}, author={Dormand, JR and Prince, PJ}, journal={Computers \&amp; Mathematics with Applications}, volume={13}, number={12}, pages={937–949}, year={1987}, publisher={Elsevier} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L167-L184">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN6FM" href="#OrdinaryDiffEqRKN.DPRKN6FM"><code>OrdinaryDiffEqRKN.DPRKN6FM</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN6FM</code></pre><p>6th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Compared to <a href="#OrdinaryDiffEqRKN.DPRKN6"><code>DPRKN6</code></a>, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L187-L205">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN8" href="#OrdinaryDiffEqRKN.DPRKN8"><code>OrdinaryDiffEqRKN.DPRKN8</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN8</code></pre><p>8th order explicit Runge-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Not as efficient as <a href="#OrdinaryDiffEqRKN.DPRKN12"><code>DPRKN12</code></a> when high accuracy is needed, however this solver is competitive with <a href="#OrdinaryDiffEqRKN.DPRKN6"><code>DPRKN6</code></a> at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L208-L228">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.DPRKN12" href="#OrdinaryDiffEqRKN.DPRKN12"><code>OrdinaryDiffEqRKN.DPRKN12</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DPRKN12</code></pre><p>12th order explicit Rugne-Kutta-Nyström method. The second order ODE should not depend on the first derivative.</p><p>Most efficient when high accuracy is needed.</p><p><strong>References</strong></p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L231-L250">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN4" href="#OrdinaryDiffEqRKN.ERKN4"><code>OrdinaryDiffEqRKN.ERKN4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN4</code></pre><p>Embedded 4(3) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic problems.</p><p><strong>References</strong></p><p>@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{&quot;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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L253-L273">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN5" href="#OrdinaryDiffEqRKN.ERKN5"><code>OrdinaryDiffEqRKN.ERKN5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN5</code></pre><p>Embedded 5(4) pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic problems.</p><p><strong>References</strong></p><p>@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{&quot;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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L276-L297">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqRKN.ERKN7" href="#OrdinaryDiffEqRKN.ERKN7"><code>OrdinaryDiffEqRKN.ERKN7</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ERKN7</code></pre><p>Embedded pair of explicit Runge-Kutta-Nyström methods. Integrates the periodic properties of the harmonic oscillator exactly.</p><p>The second order ODE should not depend on the first derivative.</p><p>Uses adaptive step size control. This method is extra efficient on periodic Problems.</p><p><strong>References</strong></p><p>@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{&quot;o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqRKN/src/algorithms.jl#L300-L318">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../stiff/implicit_extrapolation/">« Implicit Extrapolation Methods</a><a class="docs-footer-nextpage" href="../symplectic/">Symplectic Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSymplecticRK.SofSpa10" href="#OrdinaryDiffEqSymplecticRK.SofSpa10"><code>OrdinaryDiffEqSymplecticRK.SofSpa10</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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 \&amp; Francis} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSymplecticRK/src/algorithms.jl#L212-L223">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../nystrom/">« Runge-Kutta Nystrom Methods</a><a class="docs-footer-nextpage" href="../../imex/imex_multistep/">IMEX Multistep Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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class="admonition-body"><p>Missing docstring for <code>CNAB2</code>. 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Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF2" href="#OrdinaryDiffEqBDF.SBDF2"><code>OrdinaryDiffEqBDF.SBDF2</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF2(;kwargs...)</code></pre><p>The two-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L87-L99">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF3" href="#OrdinaryDiffEqBDF.SBDF3"><code>OrdinaryDiffEqBDF.SBDF3</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF3(;kwargs...)</code></pre><p>The three-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L102-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF4" href="#OrdinaryDiffEqBDF.SBDF4"><code>OrdinaryDiffEqBDF.SBDF4</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF4(;kwargs...)</code></pre><p>The four-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L117-L129">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dynamical/symplectic/">« Symplectic Runge-Kutta Methods</a><a class="docs-footer-nextpage" href="../imex_sdirk/">IMEX SDIRK Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/imex/imex_multistep.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="IMEX-Multistep-Methods"><a class="docs-heading-anchor" href="#IMEX-Multistep-Methods">IMEX Multistep Methods</a><a id="IMEX-Multistep-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#IMEX-Multistep-Methods" title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>CNAB2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>CNLF2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SBDF</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF2" href="#OrdinaryDiffEqBDF.SBDF2"><code>OrdinaryDiffEqBDF.SBDF2</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF2(;kwargs...)</code></pre><p>The two-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L87-L99">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF3" href="#OrdinaryDiffEqBDF.SBDF3"><code>OrdinaryDiffEqBDF.SBDF3</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF3(;kwargs...)</code></pre><p>The three-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L102-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.SBDF4" href="#OrdinaryDiffEqBDF.SBDF4"><code>OrdinaryDiffEqBDF.SBDF4</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">SBDF4(;kwargs...)</code></pre><p>The four-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>See also <code>SBDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L117-L129">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dynamical/symplectic/">« Symplectic Runge-Kutta Methods</a><a class="docs-footer-nextpage" href="../imex_sdirk/">IMEX SDIRK Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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Solvers</span><ul><li><a class="tocitem" href="../../misc/">-</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">IMEX Solvers</a></li><li class="is-active"><a href>IMEX SDIRK Methods</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>IMEX SDIRK Methods</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/imex/imex_sdirk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="IMEX-SDIRK-Methods"><a class="docs-heading-anchor" href="#IMEX-SDIRK-Methods">IMEX SDIRK Methods</a><a id="IMEX-SDIRK-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#IMEX-SDIRK-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.IMEXEuler" href="#OrdinaryDiffEqBDF.IMEXEuler"><code>OrdinaryDiffEqBDF.IMEXEuler</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">IMEXEuler(;kwargs...)</code></pre><p>The one-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>When applied to a <code>SplitODEProblem</code> of the form</p><pre><code class="nohighlight hljs">u&#39;(t) = f1(u) + f2(u)</code></pre><p>The default <code>IMEXEuler()</code> method uses an update of the form</p><pre><code class="nohighlight hljs">unew = uold + dt * (f1(unew) + f2(uold))</code></pre><p>See also <code>SBDF</code>, <code>IMEXEulerARK</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L335-L359">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.IMEXEulerARK" href="#OrdinaryDiffEqBDF.IMEXEulerARK"><code>OrdinaryDiffEqBDF.IMEXEulerARK</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">IMEXEulerARK(;kwargs...)</code></pre><p>The one-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>When applied to a <code>SplitODEProblem</code> of the form</p><pre><code class="nohighlight hljs">u&#39;(t) = f1(u) + f2(u)</code></pre><p>A classical additive Runge-Kutta method in the sense of <a href="https://doi.org/10.1137/S0036142995292128">Araújo, Murua, Sanz-Serna (1997)</a> consisting of the implicit and the explicit Euler method given by</p><pre><code class="nohighlight hljs">y1   = uold + dt * f1(y1)
-unew = uold + dt * (f1(unew) + f2(y1))</code></pre><p>See also <code>SBDF</code>, <code>IMEXEuler</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L362-L389">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp3" href="#OrdinaryDiffEqSDIRK.KenCarp3"><code>OrdinaryDiffEqSDIRK.KenCarp3</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L257-L267">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp4" href="#OrdinaryDiffEqSDIRK.KenCarp4"><code>OrdinaryDiffEqSDIRK.KenCarp4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L588-L598">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp47" href="#OrdinaryDiffEqSDIRK.KenCarp47"><code>OrdinaryDiffEqSDIRK.KenCarp47</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp47: SDIRK Method An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L623-L636">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp5" href="#OrdinaryDiffEqSDIRK.KenCarp5"><code>OrdinaryDiffEqSDIRK.KenCarp5</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L658-L668">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp58" href="#OrdinaryDiffEqSDIRK.KenCarp58"><code>OrdinaryDiffEqSDIRK.KenCarp58</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp58: SDIRK Method An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L690-L703">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ESDIRK54I8L2SA</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK436L2SA2" href="#OrdinaryDiffEqSDIRK.ESDIRK436L2SA2"><code>OrdinaryDiffEqSDIRK.ESDIRK436L2SA2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L745-L754">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK437L2SA" href="#OrdinaryDiffEqSDIRK.ESDIRK437L2SA"><code>OrdinaryDiffEqSDIRK.ESDIRK437L2SA</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L774-L783">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK547L2SA2" href="#OrdinaryDiffEqSDIRK.ESDIRK547L2SA2"><code>OrdinaryDiffEqSDIRK.ESDIRK547L2SA2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L803-L812">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK659L2SA" href="#OrdinaryDiffEqSDIRK.ESDIRK659L2SA"><code>OrdinaryDiffEqSDIRK.ESDIRK659L2SA</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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}</p><p>Currently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details. }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L832-L844">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../imex_multistep/">« IMEX Multistep Methods</a><a class="docs-footer-nextpage" href="../../semilinear/exponential_rk/">Exponential Runge-Kutta Integrators »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/imex/imex_sdirk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="IMEX-SDIRK-Methods"><a class="docs-heading-anchor" href="#IMEX-SDIRK-Methods">IMEX SDIRK Methods</a><a id="IMEX-SDIRK-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#IMEX-SDIRK-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.IMEXEuler" href="#OrdinaryDiffEqBDF.IMEXEuler"><code>OrdinaryDiffEqBDF.IMEXEuler</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">IMEXEuler(;kwargs...)</code></pre><p>The one-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>When applied to a <code>SplitODEProblem</code> of the form</p><pre><code class="nohighlight hljs">u&#39;(t) = f1(u) + f2(u)</code></pre><p>The default <code>IMEXEuler()</code> method uses an update of the form</p><pre><code class="nohighlight hljs">unew = uold + dt * (f1(unew) + f2(uold))</code></pre><p>See also <code>SBDF</code>, <code>IMEXEulerARK</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L335-L359">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.IMEXEulerARK" href="#OrdinaryDiffEqBDF.IMEXEulerARK"><code>OrdinaryDiffEqBDF.IMEXEulerARK</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">IMEXEulerARK(;kwargs...)</code></pre><p>The one-step version of the IMEX multistep methods of</p><ul><li>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: <a href="https://doi.org/10.1137/0732037">https://doi.org/10.1137/0732037</a></li></ul><p>When applied to a <code>SplitODEProblem</code> of the form</p><pre><code class="nohighlight hljs">u&#39;(t) = f1(u) + f2(u)</code></pre><p>A classical additive Runge-Kutta method in the sense of <a href="https://doi.org/10.1137/S0036142995292128">Araújo, Murua, Sanz-Serna (1997)</a> consisting of the implicit and the explicit Euler method given by</p><pre><code class="nohighlight hljs">y1   = uold + dt * f1(y1)
+unew = uold + dt * (f1(unew) + f2(y1))</code></pre><p>See also <code>SBDF</code>, <code>IMEXEuler</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L362-L389">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp3" href="#OrdinaryDiffEqSDIRK.KenCarp3"><code>OrdinaryDiffEqSDIRK.KenCarp3</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L257-L267">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp4" href="#OrdinaryDiffEqSDIRK.KenCarp4"><code>OrdinaryDiffEqSDIRK.KenCarp4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L588-L598">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp47" href="#OrdinaryDiffEqSDIRK.KenCarp47"><code>OrdinaryDiffEqSDIRK.KenCarp47</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp47: SDIRK Method An A-L stable stiffly-accurate 4th order seven-stage ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L623-L636">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp5" href="#OrdinaryDiffEqSDIRK.KenCarp5"><code>OrdinaryDiffEqSDIRK.KenCarp5</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L658-L668">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.KenCarp58" href="#OrdinaryDiffEqSDIRK.KenCarp58"><code>OrdinaryDiffEqSDIRK.KenCarp58</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>KenCarp58: SDIRK Method An A-L stable stiffly-accurate 5th order eight-stage ESDIRK method with splitting</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L690-L703">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ESDIRK54I8L2SA</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK436L2SA2" href="#OrdinaryDiffEqSDIRK.ESDIRK436L2SA2"><code>OrdinaryDiffEqSDIRK.ESDIRK436L2SA2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L745-L754">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK437L2SA" href="#OrdinaryDiffEqSDIRK.ESDIRK437L2SA"><code>OrdinaryDiffEqSDIRK.ESDIRK437L2SA</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L774-L783">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK547L2SA2" href="#OrdinaryDiffEqSDIRK.ESDIRK547L2SA2"><code>OrdinaryDiffEqSDIRK.ESDIRK547L2SA2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L803-L812">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ESDIRK659L2SA" href="#OrdinaryDiffEqSDIRK.ESDIRK659L2SA"><code>OrdinaryDiffEqSDIRK.ESDIRK659L2SA</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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}</p><p>Currently has STABILITY ISSUES, causing it to fail the adaptive tests. Check issue https://github.com/SciML/OrdinaryDiffEq.jl/issues/1933 for more details. }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L832-L844">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../imex_multistep/">« IMEX Multistep Methods</a><a class="docs-footer-nextpage" href="../../semilinear/exponential_rk/">Exponential Runge-Kutta Integrators »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Explicit-Runge-Kutta-Methods">Explicit Runge-Kutta Methods</a><a id="Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Explicit-Runge-Kutta-Methods" title="Permalink"></a></h1><p>With the help of <a href="https://github.com/YingboMa/FastBroadcast.jl">FastBroadcast.jl</a>, we can use threaded parallelism to reduce compute time for all of the explicit Runge-Kutta methods! The <code>thread</code> option determines whether internal broadcasting on appropriate CPU arrays should be serial (<code>thread = OrdinaryDiffEq.False()</code>, default) or use multiple threads (<code>thread = OrdinaryDiffEq.True()</code>) when Julia is started with multiple threads. When we call <code>solve(prob, alg(thread=OrdinaryDiffEq.True()))</code>, we can turn on the multithreading option to achieve acceleration (for sufficiently large problems).</p><h2 id="Standard-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Standard-Explicit-Runge-Kutta-Methods">Standard Explicit Runge-Kutta Methods</a><a id="Standard-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Heun</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Ralston</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Midpoint</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RK4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RKM</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>MSRK5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>MSRK6</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Anas5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RKO65</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>BS3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>DP5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Tsit5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>DP8</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>TanYam7</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>TsitPap8</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin10" href="#OrdinaryDiffEqFeagin.Feagin10"><code>OrdinaryDiffEqFeagin.Feagin10</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \&amp; Scientific Computations} }</p><p>Feagin10: Explicit Runge-Kutta Method Feagin&#39;s 10th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L1-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin12" href="#OrdinaryDiffEqFeagin.Feagin12"><code>OrdinaryDiffEqFeagin.Feagin12</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \&amp; Scientific Computations} }</p><p>Feagin12: Explicit Runge-Kutta Method Feagin&#39;s 12th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L16-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin14" href="#OrdinaryDiffEqFeagin.Feagin14"><code>OrdinaryDiffEqFeagin.Feagin14</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Feagin, T., “An Explicit Runge-Kutta Method of Order Fourteen,” Numerical Algorithms, 2009</p><p>Feagin14: Explicit Runge-Kutta Method Feagin&#39;s 14th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L31-L37">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>FRK65</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>PFRK87</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Stepanov5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SIR54</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina6</code>. Check Documenter&#39;s build log for details.</p></div></div><h2 id="Lazy-Interpolation-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Lazy-Interpolation-Explicit-Runge-Kutta-Methods">Lazy Interpolation Explicit Runge-Kutta Methods</a><a id="Lazy-Interpolation-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-Interpolation-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>BS5</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern6" href="#OrdinaryDiffEqVerner.Vern6"><code>OrdinaryDiffEqVerner.Vern6</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+</script><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script 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href="#Standard-Explicit-Runge-Kutta-Methods"><span>Standard Explicit Runge-Kutta Methods</span></a></li><li><a class="tocitem" href="#Lazy-Interpolation-Explicit-Runge-Kutta-Methods"><span>Lazy Interpolation Explicit Runge-Kutta Methods</span></a></li><li><a class="tocitem" href="#Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods"><span>Fixed Timestep Only Explicit Runge-Kutta Methods</span></a></li><li><a class="tocitem" href="#Parallel-Explicit-Runge-Kutta-Methods"><span>Parallel Explicit Runge-Kutta Methods</span></a></li></ul></li><li><a class="tocitem" href="../lowstorage_ssprk/">PDE-Specialized Explicit Runge-Kutta Methods</a></li><li><a class="tocitem" href="../explicit_extrapolation/">Explicit Extrapolation Methods</a></li><li><a class="tocitem" href="../nonstiff_multistep/">Multistep Methods for Non-Stiff Equations</a></li></ul></li><li><span class="tocitem">Standard Stiff ODEProblem Solvers</span><ul><li><a class="tocitem" href="../../stiff/firk/">Fully Implicit Runge-Kutta (FIRK) Methods</a></li><li><a class="tocitem" href="../../stiff/rosenbrock/">Rosenbrock Methods</a></li><li><a class="tocitem" href="../../stiff/stabilized_rk/">Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a></li><li><a class="tocitem" href="../../stiff/sdirk/">Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods</a></li><li><a class="tocitem" href="../../stiff/stiff_multistep/">Multistep Methods for Stiff Equations</a></li><li><a class="tocitem" href="../../stiff/implicit_extrapolation/">Implicit Extrapolation Methods</a></li></ul></li><li><span class="tocitem">Second Order and Dynamical ODE Solvers</span><ul><li><a class="tocitem" href="../../dynamical/nystrom/">Runge-Kutta Nystrom Methods</a></li><li><a class="tocitem" href="../../dynamical/symplectic/">Symplectic Runge-Kutta Methods</a></li></ul></li><li><span class="tocitem">IMEX Solvers</span><ul><li><a class="tocitem" href="../../imex/imex_multistep/">IMEX Multistep Methods</a></li><li><a class="tocitem" href="../../imex/imex_sdirk/">IMEX SDIRK Methods</a></li></ul></li><li><span class="tocitem">Semilinear ODE Solvers</span><ul><li><a class="tocitem" href="../../semilinear/exponential_rk/">Exponential Runge-Kutta Integrators</a></li><li><a class="tocitem" href="../../semilinear/magnus/">Magnus and Lie Group Integrators</a></li></ul></li><li><span class="tocitem">DAEProblem Solvers</span><ul><li><a class="tocitem" href="../../dae/fully_implicit/">Methods for Fully Implicit ODEs (DAEProblem)</a></li></ul></li><li><span class="tocitem">Misc Solvers</span><ul><li><a class="tocitem" href="../../misc/">-</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Standard Non-Stiff ODEProblem Solvers</a></li><li class="is-active"><a href>Explicit Runge-Kutta Methods</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Explicit Runge-Kutta Methods</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/nonstiff/explicitrk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Explicit-Runge-Kutta-Methods">Explicit Runge-Kutta Methods</a><a id="Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Explicit-Runge-Kutta-Methods" title="Permalink"></a></h1><p>With the help of <a href="https://github.com/YingboMa/FastBroadcast.jl">FastBroadcast.jl</a>, we can use threaded parallelism to reduce compute time for all of the explicit Runge-Kutta methods! The <code>thread</code> option determines whether internal broadcasting on appropriate CPU arrays should be serial (<code>thread = OrdinaryDiffEq.False()</code>, default) or use multiple threads (<code>thread = OrdinaryDiffEq.True()</code>) when Julia is started with multiple threads. When we call <code>solve(prob, alg(thread=OrdinaryDiffEq.True()))</code>, we can turn on the multithreading option to achieve acceleration (for sufficiently large problems).</p><h2 id="Standard-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Standard-Explicit-Runge-Kutta-Methods">Standard Explicit Runge-Kutta Methods</a><a id="Standard-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Heun</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Ralston</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Midpoint</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RK4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RKM</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>MSRK5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>MSRK6</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Anas5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RKO65</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>OwrenZen5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>BS3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>DP5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Tsit5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>DP8</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>TanYam7</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>TsitPap8</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin10" href="#OrdinaryDiffEqFeagin.Feagin10"><code>OrdinaryDiffEqFeagin.Feagin10</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \&amp; Scientific Computations} }</p><p>Feagin10: Explicit Runge-Kutta Method Feagin&#39;s 10th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L1-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin12" href="#OrdinaryDiffEqFeagin.Feagin12"><code>OrdinaryDiffEqFeagin.Feagin12</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@article{feagin2012high, title={High-order explicit Runge-Kutta methods using m-symmetry}, author={Feagin, Terry}, year={2012}, publisher={Neural, Parallel \&amp; Scientific Computations} }</p><p>Feagin12: Explicit Runge-Kutta Method Feagin&#39;s 12th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L16-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFeagin.Feagin14" href="#OrdinaryDiffEqFeagin.Feagin14"><code>OrdinaryDiffEqFeagin.Feagin14</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Feagin, T., “An Explicit Runge-Kutta Method of Order Fourteen,” Numerical Algorithms, 2009</p><p>Feagin14: Explicit Runge-Kutta Method Feagin&#39;s 14th-order Runge-Kutta method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqFeagin/src/algorithms.jl#L31-L37">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>FRK65</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>PFRK87</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Stepanov5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SIR54</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Alshina6</code>. Check Documenter&#39;s build log for details.</p></div></div><h2 id="Lazy-Interpolation-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Lazy-Interpolation-Explicit-Runge-Kutta-Methods">Lazy Interpolation Explicit Runge-Kutta Methods</a><a id="Lazy-Interpolation-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-Interpolation-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>BS5</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern6" href="#OrdinaryDiffEqVerner.Vern6"><code>OrdinaryDiffEqVerner.Vern6</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern6(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 6/5 Runge-Kutta method. (lazy 6th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L1-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern7" href="#OrdinaryDiffEqVerner.Vern7"><code>OrdinaryDiffEqVerner.Vern7</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 6/5 Runge-Kutta method. (lazy 6th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L1-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern7" href="#OrdinaryDiffEqVerner.Vern7"><code>OrdinaryDiffEqVerner.Vern7</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern7(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 7/6 Runge-Kutta method. (lazy 7th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L30-L55">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern8" href="#OrdinaryDiffEqVerner.Vern8"><code>OrdinaryDiffEqVerner.Vern8</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 7/6 Runge-Kutta method. (lazy 7th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L30-L55">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern8" href="#OrdinaryDiffEqVerner.Vern8"><code>OrdinaryDiffEqVerner.Vern8</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern8(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 8/7 Runge-Kutta method. (lazy 8th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L59-L84">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern9" href="#OrdinaryDiffEqVerner.Vern9"><code>OrdinaryDiffEqVerner.Vern9</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern9(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 8/7 Runge-Kutta method. (lazy 8th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L59-L84">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqVerner.Vern9" href="#OrdinaryDiffEqVerner.Vern9"><code>OrdinaryDiffEqVerner.Vern9</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Vern9(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
         step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 9/8 Runge-Kutta method. (lazy9th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L88-L113">source</a></section></article><h2 id="Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods">Fixed Timestep Only Explicit Runge-Kutta Methods</a><a id="Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Euler</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RK46NL" href="#OrdinaryDiffEqLowStorageRK.RK46NL"><code>OrdinaryDiffEqLowStorageRK.RK46NL</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RK46NL(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-         step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Julien Berland, Christophe Bogey, Christophe Bailly. Low-Dissipation and Low-Dispersion Fourth-Order Runge-Kutta Algorithm. Computers &amp; Fluids, 35(10), pp 1459-1463, 2006. doi: https://doi.org/10.1016/j.compfluid.2005.04.003</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L252-L267">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ORK256" href="#OrdinaryDiffEqLowStorageRK.ORK256"><code>OrdinaryDiffEqLowStorageRK.ORK256</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ORK256(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+        lazy = true)</code></pre><p>Explicit Runge-Kutta Method.  Verner&#39;s “Most Efficient” 9/8 Runge-Kutta method. (lazy9th order interpolant).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>lazy</code>: determines if the lazy interpolant is used.</li></ul><p><strong>References</strong></p><p>@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}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqVerner/src/algorithms.jl#L88-L113">source</a></section></article><h2 id="Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods">Fixed Timestep Only Explicit Runge-Kutta Methods</a><a id="Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed-Timestep-Only-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Euler</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RK46NL" href="#OrdinaryDiffEqLowStorageRK.RK46NL"><code>OrdinaryDiffEqLowStorageRK.RK46NL</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RK46NL(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+         step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Julien Berland, Christophe Bogey, Christophe Bailly. Low-Dissipation and Low-Dispersion Fourth-Order Runge-Kutta Algorithm. Computers &amp; Fluids, 35(10), pp 1459-1463, 2006. doi: https://doi.org/10.1016/j.compfluid.2005.04.003</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L252-L267">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ORK256" href="#OrdinaryDiffEqLowStorageRK.ORK256"><code>OrdinaryDiffEqLowStorageRK.ORK256</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ORK256(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
          step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-         williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  A second-order, five-stage explicit Runge-Kutta method for wave propagation equations. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L1-L22">source</a></section></article><h2 id="Parallel-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Parallel-Explicit-Runge-Kutta-Methods">Parallel Explicit Runge-Kutta Methods</a><a id="Parallel-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Parallel-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>KuttaPRK2p5</code>. Check Documenter&#39;s build log for details.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../usage/">« Usage</a><a class="docs-footer-nextpage" href="../lowstorage_ssprk/">PDE-Specialized Explicit Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+         williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  A second-order, five-stage explicit Runge-Kutta method for wave propagation equations. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L1-L22">source</a></section></article><h2 id="Parallel-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Parallel-Explicit-Runge-Kutta-Methods">Parallel Explicit Runge-Kutta Methods</a><a id="Parallel-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Parallel-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>KuttaPRK2p5</code>. Check Documenter&#39;s build log for details.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../usage/">« Usage</a><a class="docs-footer-nextpage" href="../lowstorage_ssprk/">PDE-Specialized Explicit Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-heading-anchor" href="#PDE-Specialized-Explicit-Runge-Kutta-Methods">PDE-Specialized Explicit Runge-Kutta Methods</a><a id="PDE-Specialized-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#PDE-Specialized-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h1><h2 id="Low-Storage-Explicit-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Low-Storage-Explicit-Runge-Kutta-Methods">Low Storage Explicit Runge-Kutta Methods</a><a id="Low-Storage-Explicit-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Low-Storage-Explicit-Runge-Kutta-Methods" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54" href="#OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54"><code>OrdinaryDiffEqLowStorageRK.CarpenterKennedy2N54</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CarpenterKennedy2N54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
                        step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                       williamson_condition = true)</code></pre><p>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).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>@article{carpenter1994fourth,     title={Fourth-order 2N-storage Runge-Kutta schemes},     author={Carpenter, Mark H and Kennedy, Christopher A},     year={1994}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L54-L76">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.SHLDDRK64" href="#OrdinaryDiffEqLowStorageRK.SHLDDRK64"><code>OrdinaryDiffEqLowStorageRK.SHLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                       williamson_condition = true)</code></pre><p>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).</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>@article{carpenter1994fourth,     title={Fourth-order 2N-storage Runge-Kutta schemes},     author={Carpenter, Mark H and Kennedy, Christopher A},     year={1994}     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L54-L76">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.SHLDDRK64" href="#OrdinaryDiffEqLowStorageRK.SHLDDRK64"><code>OrdinaryDiffEqLowStorageRK.SHLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, six-stage explicit low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L227-L248">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SHLDDRK52" href="#OrdinaryDiffEqSSPRK.SHLDDRK52"><code>OrdinaryDiffEqSSPRK.SHLDDRK52</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK52(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L350-L364">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SHLDDRK_2N" href="#OrdinaryDiffEqSSPRK.SHLDDRK_2N"><code>OrdinaryDiffEqSSPRK.SHLDDRK_2N</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK_2N(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L323-L337">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.HSLDDRK64" href="#OrdinaryDiffEqLowStorageRK.HSLDDRK64"><code>OrdinaryDiffEqLowStorageRK.HSLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HSLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, six-stage explicit low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L227-L248">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SHLDDRK52" href="#OrdinaryDiffEqSSPRK.SHLDDRK52"><code>OrdinaryDiffEqSSPRK.SHLDDRK52</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK52(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L350-L364">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SHLDDRK_2N" href="#OrdinaryDiffEqSSPRK.SHLDDRK_2N"><code>OrdinaryDiffEqSSPRK.SHLDDRK_2N</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SHLDDRK_2N(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L323-L337">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.HSLDDRK64" href="#OrdinaryDiffEqLowStorageRK.HSLDDRK64"><code>OrdinaryDiffEqLowStorageRK.HSLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HSLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
             step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L841-L864">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK73_C" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK73_C"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK73_C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK73_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 6-stage, fourth order low-stage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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     }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L841-L864">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK73_C" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK73_C"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK73_C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK73_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L25-L48">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK84_C" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK84_C"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK84_C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK84_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L25-L48">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK84_C" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK84_C"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK84_C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK84_C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L169-L192">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK84_F" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK84_F"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK84_F</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK84_F(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L169-L192">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.DGLDDRK84_F" href="#OrdinaryDiffEqLowStorageRK.DGLDDRK84_F"><code>OrdinaryDiffEqLowStorageRK.DGLDDRK84_F</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DGLDDRK84_F(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
               step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L198-L221">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK124" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK124"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK124</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK124(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              williamson_condition = true)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L198-L221">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK124" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK124"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK124</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK124(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  12-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L81-L101">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK134" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK134"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK134</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK134(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  12-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L81-L101">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK134" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK134"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK134</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK134(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  13-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L866-L886">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK144" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK144"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK144</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK144(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  13-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L866-L886">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.NDBLSRK144" href="#OrdinaryDiffEqLowStorageRK.NDBLSRK144"><code>OrdinaryDiffEqLowStorageRK.NDBLSRK144</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NDBLSRK144(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
              step_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  14-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L106-L126">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CFRLDDRK64" href="#OrdinaryDiffEqLowStorageRK.CFRLDDRK64"><code>OrdinaryDiffEqLowStorageRK.CFRLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CFRLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 6-stage, fourth order low-storage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L131-L149">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.TSLDDRK74" href="#OrdinaryDiffEqLowStorageRK.TSLDDRK74"><code>OrdinaryDiffEqLowStorageRK.TSLDDRK74</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TSLDDRK74(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L149-L167">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK43_2" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK43_2"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK43_2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK43_2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 4-stage, third order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L442-L458">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L459-L475">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4S(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L476-L492">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L493-L509">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4M(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L510-L526">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L527-L543">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L544-L560">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3N_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L561-L577">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L578-L594">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L595-L611">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4P_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L612-L628">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3N_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L629-L645">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L646-L662">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK65_4M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  6-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L663-L677">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4FM_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                   step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L679-L694">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK75_4M_5R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  CKLLSRK75<em>4M</em>5R: Low-Storage Method 7-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L696-L711">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L266-L284">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S82(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L288-L306">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L310-L328">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S173(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L332-L350">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S94(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L354-L372">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S184(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L376-L394">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S105(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L398-L416">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S205(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L420-L438">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp35" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp35"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp35</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L713-L731">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L733-L752">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp49" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp49"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp49</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L755-L773">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L775-L794">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp510" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp510"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp510</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L797-L815">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L817-L836">source</a></section></article><h2 id="SSP-Optimized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#SSP-Optimized-Runge-Kutta-Methods">SSP Optimized Runge-Kutta Methods</a><a id="SSP-Optimized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#SSP-Optimized-Runge-Kutta-Methods" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2" href="#OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2"><code>OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">KYK2014DGSSPRK_3S2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-                     step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L893-L907">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK22" href="#OrdinaryDiffEqSSPRK.SSPRK22"><code>OrdinaryDiffEqSSPRK.SSPRK22</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK22(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A second-order, two-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L20-L38">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK33" href="#OrdinaryDiffEqSSPRK.SSPRK33"><code>OrdinaryDiffEqSSPRK.SSPRK33</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK33(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, three-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L304-L322">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53" href="#OrdinaryDiffEqSSPRK.SSPRK53"><code>OrdinaryDiffEqSSPRK.SSPRK53</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L39-L56">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.KYKSSPRK42" href="#OrdinaryDiffEqSSPRK.KYKSSPRK42"><code>OrdinaryDiffEqSSPRK.KYKSSPRK42</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">KYKSSPRK42(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L336-L350">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_2N1" href="#OrdinaryDiffEqSSPRK.SSPRK53_2N1"><code>OrdinaryDiffEqSSPRK.SSPRK53_2N1</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_2N1(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L187-L204">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_2N2" href="#OrdinaryDiffEqSSPRK.SSPRK53_2N2"><code>OrdinaryDiffEqSSPRK.SSPRK53_2N2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_2N2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L1-L18">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_H" href="#OrdinaryDiffEqSSPRK.SSPRK53_H"><code>OrdinaryDiffEqSSPRK.SSPRK53_H</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_H(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L286-L303">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK63" href="#OrdinaryDiffEqSSPRK.SSPRK63"><code>OrdinaryDiffEqSSPRK.SSPRK63</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK63(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, six-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L57-L74">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK73" href="#OrdinaryDiffEqSSPRK.SSPRK73"><code>OrdinaryDiffEqSSPRK.SSPRK73</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK73(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, seven-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L268-L285">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK83" href="#OrdinaryDiffEqSSPRK.SSPRK83"><code>OrdinaryDiffEqSSPRK.SSPRK83</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK83(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, eight-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L75-L92">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK43" href="#OrdinaryDiffEqSSPRK.SSPRK43"><code>OrdinaryDiffEqSSPRK.SSPRK43</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, four-stage explicit strong stability preserving (SSP) method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Optimal third-order explicit SSP method with four stages discovered by</p><ul><li>J. F. B. M. Kraaijevanger. &quot;Contractivity of Runge-Kutta methods.&quot; In: BIT Numerical Mathematics 31.3 (1991), pp. 482–528. <a href="https://doi.org/10.1007/BF01933264">DOI: 10.1007/BF01933264</a>.</li></ul><p>Embedded method constructed by</p><ul><li>Sidafa Conde, Imre Fekete, John N. Shadid. &quot;Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods.&quot; <a href="https://arXiv.org/abs/1806.08693">arXiv: 1806.08693</a></li></ul><p>Efficient implementation (and optimized controller) developed by</p><ul><li>Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L93-L126">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK432" href="#OrdinaryDiffEqSSPRK.SSPRK432"><code>OrdinaryDiffEqSSPRK.SSPRK432</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK432(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, four-stage explicit strong stability preserving (SSP) method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu.     Strong stability preserving Runge-Kutta and multistep time discretizations.     World Scientific, 2011.     Example 6.1</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L128-L145">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRKMSVS43" href="#OrdinaryDiffEqSSPRK.SSPRKMSVS43"><code>OrdinaryDiffEqSSPRK.SSPRKMSVS43</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRKMSVS43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Shu, Chi-Wang.     Total-variation-diminishing time discretizations.     SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084.     <a href="https://doi.org/10.1137/0909073">DOI: 10.1137/0909073</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L246-L265">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRKMSVS32" href="#OrdinaryDiffEqSSPRK.SSPRKMSVS32"><code>OrdinaryDiffEqSSPRK.SSPRKMSVS32</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRKMSVS32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Shu, Chi-Wang.     Total-variation-diminishing time discretizations.     SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084.     <a href="https://doi.org/10.1137/0909073">DOI: 10.1137/0909073</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L147-L166">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK932" href="#OrdinaryDiffEqSSPRK.SSPRK932"><code>OrdinaryDiffEqSSPRK.SSPRK932</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK932(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, nine-stage explicit strong stability preserving (SSP) method.</p><p>Consider using <code>SSPRK43</code> instead, which uses the same main method and an improved embedded method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu.     Strong stability preserving Runge-Kutta and multistep time discretizations.     World Scientific, 2011.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L225-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK54" href="#OrdinaryDiffEqSSPRK.SSPRK54"><code>OrdinaryDiffEqSSPRK.SSPRK54</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L169-L186">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK104" href="#OrdinaryDiffEqSSPRK.SSPRK104"><code>OrdinaryDiffEqSSPRK.SSPRK104</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK104(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
-           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, ten-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L206-L224">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../explicitrk/">« Explicit Runge-Kutta Methods</a><a class="docs-footer-nextpage" href="../explicit_extrapolation/">Explicit Extrapolation Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+             williamson_condition = true)</code></pre><p>Explicit Runge-Kutta Method.  14-stage, fourth order low-storage method with optimized stability regions for advection-dominated problems. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>williamson_condition</code>: allows for an optimization that allows fusing broadcast expressions with the function call <code>f</code>. However, it only works for <code>Array</code> types.</li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L106-L126">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CFRLDDRK64" href="#OrdinaryDiffEqLowStorageRK.CFRLDDRK64"><code>OrdinaryDiffEqLowStorageRK.CFRLDDRK64</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CFRLDDRK64(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 6-stage, fourth order low-storage, low-dissipation, low-dispersion scheme. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L131-L149">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.TSLDDRK74" href="#OrdinaryDiffEqLowStorageRK.TSLDDRK74"><code>OrdinaryDiffEqLowStorageRK.TSLDDRK74</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TSLDDRK74(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L149-L167">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK43_2" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK43_2"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK43_2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK43_2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 4-stage, third order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L442-L458">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L459-L475">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4S</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4S(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L476-L492">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4C</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4C(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L493-L509">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK95_4M</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK95_4M(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+               step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 9-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L510-L526">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3C_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L527-L543">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L544-L560">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3N_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L561-L577">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4C_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4C_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L578-L594">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4M_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4M_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L595-L611">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4P_3R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4P_3R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L612-L628">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3N_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3N_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L629-L645">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK54_3M_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK54_3M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 5-stage, fourth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L646-L662">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK65_4M_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK65_4M_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  6-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L663-L677">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK85_4FM_4R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK85_4FM_4R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                   step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  Low-Storage Method 8-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L679-L694">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R" href="#OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R"><code>OrdinaryDiffEqLowStorageRK.CKLLSRK75_4M_5R</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CKLLSRK75_4M_5R(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                  step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  CKLLSRK75<em>4M</em>5R: Low-Storage Method 7-stage, fifth order low-storage scheme, optimized for compressible Navier–Stokes equations.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L696-L711">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S32</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L266-L284">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S82</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S82(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L288-L306">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S53</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L310-L328">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S173</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S173(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L332-L350">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S94</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S94(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L354-L372">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S184</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S184(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L376-L394">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S105</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S105(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L398-L416">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205" href="#OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205"><code>OrdinaryDiffEqLowStorageRK.ParsaniKetchesonDeconinck3S205</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParsaniKetchesonDeconinck3S205(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L420-L438">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp35" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp35"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp35</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L713-L731">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL35</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL35(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L733-L752">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp49" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp49"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp49</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L755-L773">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL49</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL49(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L775-L794">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3Sp510" href="#OrdinaryDiffEqLowStorageRK.RDPK3Sp510"><code>OrdinaryDiffEqLowStorageRK.RDPK3Sp510</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3Sp510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L797-L815">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510" href="#OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510"><code>OrdinaryDiffEqLowStorageRK.RDPK3SpFSAL510</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RDPK3SpFSAL510(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                 step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ranocha, Dalcin, Parsani, Ketcheson (2021)     Optimized Runge-Kutta Methods with Automatic Step Size Control for     Compressible Computational Fluid Dynamics     <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L817-L836">source</a></section></article><h2 id="SSP-Optimized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#SSP-Optimized-Runge-Kutta-Methods">SSP Optimized Runge-Kutta Methods</a><a id="SSP-Optimized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#SSP-Optimized-Runge-Kutta-Methods" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2" href="#OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2"><code>OrdinaryDiffEqLowStorageRK.KYK2014DGSSPRK_3S2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">KYK2014DGSSPRK_3S2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+                     step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqLowStorageRK/src/algorithms.jl#L893-L907">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK22" href="#OrdinaryDiffEqSSPRK.SSPRK22"><code>OrdinaryDiffEqSSPRK.SSPRK22</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK22(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A second-order, two-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L20-L38">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK33" href="#OrdinaryDiffEqSSPRK.SSPRK33"><code>OrdinaryDiffEqSSPRK.SSPRK33</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK33(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, three-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L304-L322">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53" href="#OrdinaryDiffEqSSPRK.SSPRK53"><code>OrdinaryDiffEqSSPRK.SSPRK53</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L39-L56">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.KYKSSPRK42" href="#OrdinaryDiffEqSSPRK.KYKSSPRK42"><code>OrdinaryDiffEqSSPRK.KYKSSPRK42</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">KYKSSPRK42(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+             step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  TBD</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L336-L350">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_2N1" href="#OrdinaryDiffEqSSPRK.SSPRK53_2N1"><code>OrdinaryDiffEqSSPRK.SSPRK53_2N1</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_2N1(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L187-L204">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_2N2" href="#OrdinaryDiffEqSSPRK.SSPRK53_2N2"><code>OrdinaryDiffEqSSPRK.SSPRK53_2N2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_2N2(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L1-L18">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK53_H" href="#OrdinaryDiffEqSSPRK.SSPRK53_H"><code>OrdinaryDiffEqSSPRK.SSPRK53_H</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK53_H(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+            step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, five-stage explicit strong stability preserving (SSP) low-storage method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Higueras and T. Roldán.     New third order low-storage SSP explicit Runge–Kutta methods     arXiv:1809.04807v1.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L286-L303">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK63" href="#OrdinaryDiffEqSSPRK.SSPRK63"><code>OrdinaryDiffEqSSPRK.SSPRK63</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK63(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, six-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L57-L74">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK73" href="#OrdinaryDiffEqSSPRK.SSPRK73"><code>OrdinaryDiffEqSSPRK.SSPRK73</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK73(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, seven-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L268-L285">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK83" href="#OrdinaryDiffEqSSPRK.SSPRK83"><code>OrdinaryDiffEqSSPRK.SSPRK83</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK83(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, eight-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L75-L92">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK43" href="#OrdinaryDiffEqSSPRK.SSPRK43"><code>OrdinaryDiffEqSSPRK.SSPRK43</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, four-stage explicit strong stability preserving (SSP) method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Optimal third-order explicit SSP method with four stages discovered by</p><ul><li>J. F. B. M. Kraaijevanger. &quot;Contractivity of Runge-Kutta methods.&quot; In: BIT Numerical Mathematics 31.3 (1991), pp. 482–528. <a href="https://doi.org/10.1007/BF01933264">DOI: 10.1007/BF01933264</a>.</li></ul><p>Embedded method constructed by</p><ul><li>Sidafa Conde, Imre Fekete, John N. Shadid. &quot;Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods.&quot; <a href="https://arXiv.org/abs/1806.08693">arXiv: 1806.08693</a></li></ul><p>Efficient implementation (and optimized controller) developed by</p><ul><li>Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2021) Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics <a href="https://arxiv.org/abs/2104.06836">arXiv:2104.06836</a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L93-L126">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK432" href="#OrdinaryDiffEqSSPRK.SSPRK432"><code>OrdinaryDiffEqSSPRK.SSPRK432</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK432(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, four-stage explicit strong stability preserving (SSP) method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu.     Strong stability preserving Runge-Kutta and multistep time discretizations.     World Scientific, 2011.     Example 6.1</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L128-L145">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRKMSVS43" href="#OrdinaryDiffEqSSPRK.SSPRKMSVS43"><code>OrdinaryDiffEqSSPRK.SSPRKMSVS43</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRKMSVS43(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Shu, Chi-Wang.     Total-variation-diminishing time discretizations.     SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084.     <a href="https://doi.org/10.1137/0909073">DOI: 10.1137/0909073</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L246-L265">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRKMSVS32" href="#OrdinaryDiffEqSSPRK.SSPRKMSVS32"><code>OrdinaryDiffEqSSPRK.SSPRKMSVS32</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRKMSVS32(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+              step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>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.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Shu, Chi-Wang.     Total-variation-diminishing time discretizations.     SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073-1084.     <a href="https://doi.org/10.1137/0909073">DOI: 10.1137/0909073</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L147-L166">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK932" href="#OrdinaryDiffEqSSPRK.SSPRK932"><code>OrdinaryDiffEqSSPRK.SSPRK932</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK932(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A third-order, nine-stage explicit strong stability preserving (SSP) method.</p><p>Consider using <code>SSPRK43</code> instead, which uses the same main method and an improved embedded method.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu.     Strong stability preserving Runge-Kutta and multistep time discretizations.     World Scientific, 2011.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L225-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK54" href="#OrdinaryDiffEqSSPRK.SSPRK54"><code>OrdinaryDiffEqSSPRK.SSPRK54</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK54(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+          step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, five-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>Ruuth, Steven.     Global optimization of explicit strong-stability-preserving Runge-Kutta methods.     Mathematics of Computation 75.253 (2006): 183-207.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L169-L186">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSSPRK.SSPRK104" href="#OrdinaryDiffEqSSPRK.SSPRK104"><code>OrdinaryDiffEqSSPRK.SSPRK104</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SSPRK104(; stage_limiter! = OrdinaryDiffEq.trivial_limiter!,
+           step_limiter! = OrdinaryDiffEq.trivial_limiter!)</code></pre><p>Explicit Runge-Kutta Method.  A fourth-order, ten-stage explicit strong stability preserving (SSP) method. Fixed timestep only.</p><p><strong>Keyword Arguments</strong></p><ul><li><code>stage_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li><li><code>step_limiter!</code>: function of the form <code>limiter!(u, integrator, p, t)</code></li></ul><p><strong>References</strong></p><p>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.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSSPRK/src/algorithms.jl#L206-L224">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../explicitrk/">« Explicit Runge-Kutta Methods</a><a class="docs-footer-nextpage" href="../explicit_extrapolation/">Explicit Extrapolation Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFIRK.RadauIIA3" href="#OrdinaryDiffEqFIRK.RadauIIA3"><code>OrdinaryDiffEqFIRK.RadauIIA3</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>RadauIIA3: Fully-Implicit Runge-Kutta Method An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqFIRK/src/algorithms.jl#L3-L17">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqFIRK.RadauIIA5" href="#OrdinaryDiffEqFIRK.RadauIIA5"><code>OrdinaryDiffEqFIRK.RadauIIA5</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>RadauIIA5: Fully-Implicit Runge-Kutta Method An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.</p></div><a 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Similar to Hairer&#39;s SEULEX.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L19-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitDeuflhardExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L128-L131">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitHairerWannerExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates, following Hairer&#39;s SODEX in the adaptivity behavior.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L241-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitEulerBarycentricExtrapolation: Parallelized Implicit Extrapolation Method Euler extrapolation using Barycentric coordinates, following Hairer&#39;s SODEX in the adaptivity behavior.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L302-L305">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../stiff_multistep/">« Multistep Methods for Stiff Equations</a><a class="docs-footer-nextpage" href="../../dynamical/nystrom/">Runge-Kutta Nystrom Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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Similar to Hairer&#39;s SEULEX.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L19-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitDeuflhardExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitDeuflhardExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L128-L131">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitHairerWannerExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitHairerWannerExtrapolation: Parallelized Implicit Extrapolation Method Midpoint extrapolation using Barycentric coordinates, following Hairer&#39;s SODEX in the adaptivity behavior.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L241-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation" href="#OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation"><code>OrdinaryDiffEqExtrapolation.ImplicitEulerBarycentricExtrapolation</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitEulerBarycentricExtrapolation: Parallelized Implicit Extrapolation Method Euler extrapolation using Barycentric coordinates, following Hairer&#39;s SODEX in the adaptivity behavior.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqExtrapolation/src/algorithms.jl#L302-L305">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../stiff_multistep/">« Multistep Methods for Stiff Equations</a><a class="docs-footer-nextpage" href="../../dynamical/nystrom/">Runge-Kutta Nystrom Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3PRL</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3PRL2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas3P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RosShamp4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Veldd4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Velds4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>GRK4T</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>GRK4A</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Ros4LStab</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas42</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4P2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas5P</code>. Check Documenter&#39;s build log for details.</p></div></div><h2 id="Rosenbrock-W-Methods"><a class="docs-heading-anchor" href="#Rosenbrock-W-Methods">Rosenbrock W-Methods</a><a id="Rosenbrock-W-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Rosenbrock-W-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rosenbrock23</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rosenbrock32</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas23W</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW1a</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW1b</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PRw</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROK4a</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS2S</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RosenbrockW6S4OS</code>. 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Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3PRL</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3PRL2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS3P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas3P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RosShamp4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Veldd4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Velds4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>GRK4T</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>GRK4A</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Ros4LStab</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas42</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4P</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas4P2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas5P</code>. Check Documenter&#39;s build log for details.</p></div></div><h2 id="Rosenbrock-W-Methods"><a class="docs-heading-anchor" href="#Rosenbrock-W-Methods">Rosenbrock W-Methods</a><a id="Rosenbrock-W-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Rosenbrock-W-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rosenbrock23</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rosenbrock32</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>Rodas23W</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW1a</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW1b</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW2</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PW3</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS34PRw</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROK4a</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>ROS2S</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>RosenbrockW6S4OS</code>. Check Documenter&#39;s build log for details.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../firk/">« Fully Implicit Runge-Kutta (FIRK) Methods</a><a class="docs-footer-nextpage" href="../stabilized_rk/">Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev) »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/stiff/sdirk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods"><a class="docs-heading-anchor" href="#Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods">Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods</a><a id="Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ImplicitEuler" href="#OrdinaryDiffEqSDIRK.ImplicitEuler"><code>OrdinaryDiffEqSDIRK.ImplicitEuler</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L2-L6">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ImplicitMidpoint" href="#OrdinaryDiffEqSDIRK.ImplicitMidpoint"><code>OrdinaryDiffEqSDIRK.ImplicitMidpoint</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitMidpoint: SDIRK Method A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L28-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Trapezoid" href="#OrdinaryDiffEqSDIRK.Trapezoid"><code>OrdinaryDiffEqSDIRK.Trapezoid</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Andre Vladimirescu. 1994. The Spice Book. John Wiley &amp; Sons, Inc., New York, NY, USA.</p><p>Trapezoid: SDIRK Method A second order A-stable symmetric ESDIRK method. &quot;Almost symplectic&quot; 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).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L56-L66">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.TRBDF2" href="#OrdinaryDiffEqSDIRK.TRBDF2"><code>OrdinaryDiffEqSDIRK.TRBDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>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.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L93-L108">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.SDIRK2" href="#OrdinaryDiffEqSDIRK.SDIRK2"><code>OrdinaryDiffEqSDIRK.SDIRK2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>SDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L133-L147">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SDIRK22</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SSPSDIRK2</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno3" href="#OrdinaryDiffEqSDIRK.Kvaerno3"><code>OrdinaryDiffEqSDIRK.Kvaerno3</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L220-L234">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>CFNLIRK3</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Cash4" href="#OrdinaryDiffEqSDIRK.Cash4"><code>OrdinaryDiffEqSDIRK.Cash4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Cash4: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L310-L324">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK6</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK7</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK8</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Hairer4" href="#OrdinaryDiffEqSDIRK.Hairer4"><code>OrdinaryDiffEqSDIRK.Hairer4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)</p><p>Hairer4: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L456-L463">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Hairer42" href="#OrdinaryDiffEqSDIRK.Hairer42"><code>OrdinaryDiffEqSDIRK.Hairer42</code></a> — <span class="docstring-category">Type</span></header><section><div><p>E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)</p><p>Hairer42: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L485-L492">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno4" href="#OrdinaryDiffEqSDIRK.Kvaerno4"><code>OrdinaryDiffEqSDIRK.Kvaerno4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L514-L528">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno5" href="#OrdinaryDiffEqSDIRK.Kvaerno5"><code>OrdinaryDiffEqSDIRK.Kvaerno5</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L551-L565">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../stabilized_rk/">« Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a><a class="docs-footer-nextpage" href="../stiff_multistep/">Multistep Methods for Stiff Equations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/stiff/sdirk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods"><a class="docs-heading-anchor" href="#Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods">Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods</a><a id="Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Singly-Diagonally-Implicit-Runge-Kutta-(SDIRK)-Methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ImplicitEuler" href="#OrdinaryDiffEqSDIRK.ImplicitEuler"><code>OrdinaryDiffEqSDIRK.ImplicitEuler</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L2-L6">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.ImplicitMidpoint" href="#OrdinaryDiffEqSDIRK.ImplicitMidpoint"><code>OrdinaryDiffEqSDIRK.ImplicitMidpoint</code></a> — <span class="docstring-category">Type</span></header><section><div><p>ImplicitMidpoint: SDIRK Method A second order A-stable symplectic and symmetric implicit solver. Good for highly stiff equations which need symplectic integration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L28-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Trapezoid" href="#OrdinaryDiffEqSDIRK.Trapezoid"><code>OrdinaryDiffEqSDIRK.Trapezoid</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Andre Vladimirescu. 1994. The Spice Book. John Wiley &amp; Sons, Inc., New York, NY, USA.</p><p>Trapezoid: SDIRK Method A second order A-stable symmetric ESDIRK method. &quot;Almost symplectic&quot; 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).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L56-L66">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.TRBDF2" href="#OrdinaryDiffEqSDIRK.TRBDF2"><code>OrdinaryDiffEqSDIRK.TRBDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>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.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L93-L108">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.SDIRK2" href="#OrdinaryDiffEqSDIRK.SDIRK2"><code>OrdinaryDiffEqSDIRK.SDIRK2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>SDIRK2: SDIRK Method An A-B-L stable 2nd order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L133-L147">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SDIRK22</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SSPSDIRK2</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno3" href="#OrdinaryDiffEqSDIRK.Kvaerno3"><code>OrdinaryDiffEqSDIRK.Kvaerno3</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno3: SDIRK Method An A-L stable stiffly-accurate 3rd order ESDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L220-L234">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>CFNLIRK3</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Cash4" href="#OrdinaryDiffEqSDIRK.Cash4"><code>OrdinaryDiffEqSDIRK.Cash4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Cash4: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L310-L324">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK4</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK5</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK6</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK7</code>. Check Documenter&#39;s build log for details.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SFSDIRK8</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Hairer4" href="#OrdinaryDiffEqSDIRK.Hairer4"><code>OrdinaryDiffEqSDIRK.Hairer4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)</p><p>Hairer4: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L456-L463">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Hairer42" href="#OrdinaryDiffEqSDIRK.Hairer42"><code>OrdinaryDiffEqSDIRK.Hairer42</code></a> — <span class="docstring-category">Type</span></header><section><div><p>E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)</p><p>Hairer42: SDIRK Method An A-L stable 4th order SDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L485-L492">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno4" href="#OrdinaryDiffEqSDIRK.Kvaerno4"><code>OrdinaryDiffEqSDIRK.Kvaerno4</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno4: SDIRK Method An A-L stable stiffly-accurate 4th order ESDIRK method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L514-L528">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqSDIRK.Kvaerno5" href="#OrdinaryDiffEqSDIRK.Kvaerno5"><code>OrdinaryDiffEqSDIRK.Kvaerno5</code></a> — <span class="docstring-category">Type</span></header><section><div><p>@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} }</p><p>Kvaerno5: SDIRK Method An A-L stable stiffly-accurate 5th order ESDIRK method</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqSDIRK/src/algorithms.jl#L551-L565">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../stabilized_rk/">« Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a><a class="docs-footer-nextpage" href="../stiff_multistep/">Multistep Methods for Stiff Equations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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class="is-active"><a href>Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/stiff/stabilized_rk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)"><a class="docs-heading-anchor" href="#Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)">Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a><a id="Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)-1"></a><a class="docs-heading-anchor-permalink" href="#Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)" title="Permalink"></a></h1><h2 id="Explicit-Stabilized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Explicit-Stabilized-Runge-Kutta-Methods">Explicit Stabilized Runge-Kutta Methods</a><a id="Explicit-Stabilized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Explicit-Stabilized-Runge-Kutta-Methods" title="Permalink"></a></h2><p>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 <code>eigen_est</code>, for example</p><pre><code class="language-julia hljs">`eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound`</code></pre><p>The methods <code>ROCK2</code> and <code>ROCK4</code> also include keyword arguments <code>min_stages</code> and <code>max_stages</code>, which specify upper and lower bounds on the adaptively chosen number of stages for stability.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ROCK2" href="#OrdinaryDiffEqStabilizedRK.ROCK2"><code>OrdinaryDiffEqStabilizedRK.ROCK2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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</p><p>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.</p><p>This method takes optional keyword arguments <code>min_stages</code>, <code>max_stages</code>, and <code>eigen_est</code>. The function <code>eigen_est</code> should be of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L1-L16">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ROCK4" href="#OrdinaryDiffEqStabilizedRK.ROCK4"><code>OrdinaryDiffEqStabilizedRK.ROCK4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ROCK4(; min_stages = 0, max_stages = 152, eigen_est = nothing)</code></pre><p>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</p><p>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.</p><p>This method takes optional keyword arguments <code>min_stages</code>, <code>max_stages</code>, and <code>eigen_est</code>. The function <code>eigen_est</code> should be of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L26-L44">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SERK2</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ESERK4" href="#OrdinaryDiffEqStabilizedRK.ESERK4"><code>OrdinaryDiffEqStabilizedRK.ESERK4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ESERK4(; eigen_est = nothing)</code></pre><p>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.</p><p>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.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L85-L102">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ESERK5" href="#OrdinaryDiffEqStabilizedRK.ESERK5"><code>OrdinaryDiffEqStabilizedRK.ESERK5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ESERK5(; eigen_est = nothing)</code></pre><p>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.</p><p>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.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L105-L122">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.RKC" href="#OrdinaryDiffEqStabilizedRK.RKC"><code>OrdinaryDiffEqStabilizedRK.RKC</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RKC(; eigen_est = nothing)</code></pre><p>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</p><p>RKC: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L65-L82">source</a></section></article><h2 id="Implicit-Stabilized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Implicit-Stabilized-Runge-Kutta-Methods">Implicit Stabilized Runge-Kutta Methods</a><a id="Implicit-Stabilized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Implicit-Stabilized-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>IRKC</code>. Check Documenter&#39;s build log for details.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../rosenbrock/">« Rosenbrock Methods</a><a class="docs-footer-nextpage" href="../sdirk/">Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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class="is-active"><a href>Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/stiff/stabilized_rk.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)"><a class="docs-heading-anchor" href="#Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)">Stabilized Runge-Kutta Methods (Runge-Kutta-Chebyshev)</a><a id="Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)-1"></a><a class="docs-heading-anchor-permalink" href="#Stabilized-Runge-Kutta-Methods-(Runge-Kutta-Chebyshev)" title="Permalink"></a></h1><h2 id="Explicit-Stabilized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Explicit-Stabilized-Runge-Kutta-Methods">Explicit Stabilized Runge-Kutta Methods</a><a id="Explicit-Stabilized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Explicit-Stabilized-Runge-Kutta-Methods" title="Permalink"></a></h2><p>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 <code>eigen_est</code>, for example</p><pre><code class="language-julia hljs">`eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound`</code></pre><p>The methods <code>ROCK2</code> and <code>ROCK4</code> also include keyword arguments <code>min_stages</code> and <code>max_stages</code>, which specify upper and lower bounds on the adaptively chosen number of stages for stability.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ROCK2" href="#OrdinaryDiffEqStabilizedRK.ROCK2"><code>OrdinaryDiffEqStabilizedRK.ROCK2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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</p><p>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.</p><p>This method takes optional keyword arguments <code>min_stages</code>, <code>max_stages</code>, and <code>eigen_est</code>. The function <code>eigen_est</code> should be of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L1-L16">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ROCK4" href="#OrdinaryDiffEqStabilizedRK.ROCK4"><code>OrdinaryDiffEqStabilizedRK.ROCK4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ROCK4(; min_stages = 0, max_stages = 152, eigen_est = nothing)</code></pre><p>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</p><p>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.</p><p>This method takes optional keyword arguments <code>min_stages</code>, <code>max_stages</code>, and <code>eigen_est</code>. The function <code>eigen_est</code> should be of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L26-L44">source</a></section></article><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>SERK2</code>. Check Documenter&#39;s build log for details.</p></div></div><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ESERK4" href="#OrdinaryDiffEqStabilizedRK.ESERK4"><code>OrdinaryDiffEqStabilizedRK.ESERK4</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ESERK4(; eigen_est = nothing)</code></pre><p>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.</p><p>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.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L85-L102">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.ESERK5" href="#OrdinaryDiffEqStabilizedRK.ESERK5"><code>OrdinaryDiffEqStabilizedRK.ESERK5</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ESERK5(; eigen_est = nothing)</code></pre><p>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.</p><p>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.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L105-L122">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqStabilizedRK.RKC" href="#OrdinaryDiffEqStabilizedRK.RKC"><code>OrdinaryDiffEqStabilizedRK.RKC</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RKC(; eigen_est = nothing)</code></pre><p>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</p><p>RKC: Stabilized Explicit Method. Second order stabilized Runge-Kutta method. Exhibits high stability for real eigenvalues.</p><p>This method takes the keyword argument <code>eigen_est</code> of the form</p><p><code>eigen_est = (integrator) -&gt; integrator.eigen_est = upper_bound</code>,</p><p>where <code>upper_bound</code> is an estimated upper bound on the spectral radius of the Jacobian matrix. If <code>eigen_est</code> is not provided, <code>upper_bound</code> will be estimated using the power iteration.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqStabilizedRK/src/algorithms.jl#L65-L82">source</a></section></article><h2 id="Implicit-Stabilized-Runge-Kutta-Methods"><a class="docs-heading-anchor" href="#Implicit-Stabilized-Runge-Kutta-Methods">Implicit Stabilized Runge-Kutta Methods</a><a id="Implicit-Stabilized-Runge-Kutta-Methods-1"></a><a class="docs-heading-anchor-permalink" href="#Implicit-Stabilized-Runge-Kutta-Methods" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Missing docstring.</header><div class="admonition-body"><p>Missing docstring for <code>IRKC</code>. Check Documenter&#39;s build log for details.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../rosenbrock/">« Rosenbrock Methods</a><a class="docs-footer-nextpage" href="../sdirk/">Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. 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GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/master/docs/src/stiff/stiff_multistep.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Multistep-Methods-for-Stiff-Equations"><a class="docs-heading-anchor" href="#Multistep-Methods-for-Stiff-Equations">Multistep Methods for Stiff Equations</a><a id="Multistep-Methods-for-Stiff-Equations-1"></a><a class="docs-heading-anchor-permalink" href="#Multistep-Methods-for-Stiff-Equations" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF1" href="#OrdinaryDiffEqBDF.QNDF1"><code>OrdinaryDiffEqBDF.QNDF1</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF1: Multistep Method An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine&#39;s accuracy-optimal -0.1850.</p><p>See also <code>QNDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L132-L138">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF1" href="#OrdinaryDiffEqBDF.QBDF1"><code>OrdinaryDiffEqBDF.QBDF1</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF1: Multistep Method</p><p>An alias of <code>QNDF1</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L314-L318">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF2" href="#OrdinaryDiffEqBDF.QNDF2"><code>OrdinaryDiffEqBDF.QNDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF2: Multistep Method An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.</p><p>See also <code>QNDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L167-L172">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF2" href="#OrdinaryDiffEqBDF.QBDF2"><code>OrdinaryDiffEqBDF.QBDF2</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF2: Multistep Method</p><p>An alias of <code>QNDF2</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L321-L325">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.ABDF2" href="#OrdinaryDiffEqBDF.ABDF2"><code>OrdinaryDiffEqBDF.ABDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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</p><p>ABDF2: Multistep Method An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L1-L8">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF" href="#OrdinaryDiffEqBDF.QNDF"><code>OrdinaryDiffEqBDF.QNDF</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF: Multistep Method An adaptive order quasi-constant timestep NDF method. Utilizes Shampine&#39;s accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).</p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L201-L216">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF" href="#OrdinaryDiffEqBDF.QBDF"><code>OrdinaryDiffEqBDF.QBDF</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF: Multistep Method</p><p>An alias of <code>QNDF</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L328-L332">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.FBDF" href="#OrdinaryDiffEqBDF.FBDF"><code>OrdinaryDiffEqBDF.FBDF</code></a> — <span class="docstring-category">Type</span></header><section><div><p>FBDF: Fixed leading coefficient BDF</p><p>An adaptive order quasi-constant timestep NDF method. Utilizes Shampine&#39;s accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).</p><p>@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH \&amp; Co. KG} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L272-L284">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.MEBDF2" href="#OrdinaryDiffEqBDF.MEBDF2"><code>OrdinaryDiffEqBDF.MEBDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>MEBDF2: Multistep Method The second order Modified Extended BDF method, which has improved stability properties over the standard BDF. Fixed timestep only.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/3ef01f5b538bf9c393dc3000777d18d74468f31c/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L248-L252">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../sdirk/">« Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods</a><a class="docs-footer-nextpage" href="../implicit_extrapolation/">Implicit Extrapolation Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. 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class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF1" href="#OrdinaryDiffEqBDF.QNDF1"><code>OrdinaryDiffEqBDF.QNDF1</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF1: Multistep Method An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine&#39;s accuracy-optimal -0.1850.</p><p>See also <code>QNDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L132-L138">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF1" href="#OrdinaryDiffEqBDF.QBDF1"><code>OrdinaryDiffEqBDF.QBDF1</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF1: Multistep Method</p><p>An alias of <code>QNDF1</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L314-L318">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF2" href="#OrdinaryDiffEqBDF.QNDF2"><code>OrdinaryDiffEqBDF.QNDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF2: Multistep Method An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.</p><p>See also <code>QNDF</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L167-L172">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF2" href="#OrdinaryDiffEqBDF.QBDF2"><code>OrdinaryDiffEqBDF.QBDF2</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF2: Multistep Method</p><p>An alias of <code>QNDF2</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L321-L325">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.ABDF2" href="#OrdinaryDiffEqBDF.ABDF2"><code>OrdinaryDiffEqBDF.ABDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>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</p><p>ABDF2: Multistep Method An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L1-L8">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QNDF" href="#OrdinaryDiffEqBDF.QNDF"><code>OrdinaryDiffEqBDF.QNDF</code></a> — <span class="docstring-category">Type</span></header><section><div><p>QNDF: Multistep Method An adaptive order quasi-constant timestep NDF method. Utilizes Shampine&#39;s accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).</p><p>@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} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L201-L216">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.QBDF" href="#OrdinaryDiffEqBDF.QBDF"><code>OrdinaryDiffEqBDF.QBDF</code></a> — <span class="docstring-category">Function</span></header><section><div><p>QBDF: Multistep Method</p><p>An alias of <code>QNDF</code> with κ=0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L328-L332">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.FBDF" href="#OrdinaryDiffEqBDF.FBDF"><code>OrdinaryDiffEqBDF.FBDF</code></a> — <span class="docstring-category">Type</span></header><section><div><p>FBDF: Fixed leading coefficient BDF</p><p>An adaptive order quasi-constant timestep NDF method. Utilizes Shampine&#39;s accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).</p><p>@article{shampine2002solving, title={Solving 0= F (t, y (t), y′(t)) in Matlab}, author={Shampine, Lawrence F}, year={2002}, publisher={Walter de Gruyter GmbH \&amp; Co. KG} }</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L272-L284">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="OrdinaryDiffEqBDF.MEBDF2" href="#OrdinaryDiffEqBDF.MEBDF2"><code>OrdinaryDiffEqBDF.MEBDF2</code></a> — <span class="docstring-category">Type</span></header><section><div><p>MEBDF2: Multistep Method The second order Modified Extended BDF method, which has improved stability properties over the standard BDF. Fixed timestep only.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/OrdinaryDiffEq.jl/blob/c5c02d6196218950eb4bbece3d7d839f43f3c258/lib/OrdinaryDiffEqBDF/src/algorithms.jl#L248-L252">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../sdirk/">« Singly-Diagonally Implicit Runge-Kutta (SDIRK) Methods</a><a class="docs-footer-nextpage" href="../implicit_extrapolation/">Implicit Extrapolation Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR2390/usage/index.html b/previews/PR2390/usage/index.html
index 5d5d64f438..176bb1abe6 100644
--- a/previews/PR2390/usage/index.html
+++ b/previews/PR2390/usage/index.html
@@ -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);</code></pre><p>Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).</p><h2 id="Available-Solvers"><a class="docs-heading-anchor" href="#Available-Solvers">Available Solvers</a><a id="Available-Solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Available-Solvers" title="Permalink"></a></h2><p>For the list of available solvers, please refer to the <a href="https://diffeq.sciml.ai/dev/solvers/ode_solve/">DifferentialEquations.jl ODE Solvers</a>, <a href="http://diffeq.sciml.ai/dev/solvers/dynamical_solve/">Dynamical ODE Solvers</a>, and the <a href="http://diffeq.sciml.ai/dev/solvers/split_ode_solve/">Split ODE Solvers</a> pages.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« OrdinaryDiffEq.jl: ODE solvers and utilities</a><a class="docs-footer-nextpage" href="../nonstiff/explicitrk/">Explicit Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 10:49">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+sol2 = solve(prob, KahanLi8(), dt = 1 / 10);</code></pre><p>Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).</p><h2 id="Available-Solvers"><a class="docs-heading-anchor" href="#Available-Solvers">Available Solvers</a><a id="Available-Solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Available-Solvers" title="Permalink"></a></h2><p>For the list of available solvers, please refer to the <a href="https://diffeq.sciml.ai/dev/solvers/ode_solve/">DifferentialEquations.jl ODE Solvers</a>, <a href="http://diffeq.sciml.ai/dev/solvers/dynamical_solve/">Dynamical ODE Solvers</a>, and the <a href="http://diffeq.sciml.ai/dev/solvers/split_ode_solve/">Split ODE Solvers</a> pages.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« OrdinaryDiffEq.jl: ODE solvers and utilities</a><a class="docs-footer-nextpage" href="../nonstiff/explicitrk/">Explicit Runge-Kutta Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 18 August 2024 11:04">Sunday 18 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>