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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Old LSRK Formulations · ClimaTimeSteppers</title><meta name="title" content="Old LSRK Formulations · ClimaTimeSteppers"/><meta property="og:title" content="Old LSRK Formulations · ClimaTimeSteppers"/><meta property="twitter:title" content="Old LSRK Formulations · ClimaTimeSteppers"/><meta name="description" content="Documentation for ClimaTimeSteppers."/><meta property="og:description" content="Documentation for ClimaTimeSteppers."/><meta property="twitter:description" content="Documentation for ClimaTimeSteppers."/><script data-outdated-warner src="../../assets/warner.js"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 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href="../../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">ClimaTimeSteppers</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">ClimaTimeSteppers.jl</a></li><li><input class="collapse-toggle" id="menuitem-2" type="checkbox" checked/><label class="tocitem" for="menuitem-2"><span class="docs-label">Algorithm Formulations</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../ode_solvers/">ODE Solvers</a></li><li><a class="tocitem" href="../newtons_method/">Newtons Method</a></li><li><a class="tocitem" href="../rosenbrock/">Rosenbrock Method</a></li><li class="is-active"><a class="tocitem" href>Old LSRK Formulations</a></li><li><a class="tocitem" href="../mrrk/">Old MRRK Formulations</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Test problems</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../test_problems/">Test problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">Tutorials</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/diffusion/">A full example on how to use ClimaTimesteppers with ClimaCore</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">API docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../api/ode_solvers/">ODE Solvers</a></li><li><a class="tocitem" href="../../api/newtons_method/">Newtons Method</a></li><li><a class="tocitem" href="../../api/callbacks/">Callbacks</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-6" type="checkbox"/><label class="tocitem" for="menuitem-6"><span class="docs-label">Developer docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../dev/types/">Types</a></li><li><a class="tocitem" href="../../dev/report_gen/">Report generator</a></li></ul></li><li><a class="tocitem" href="../../references/">References</a></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">Algorithm Formulations</a></li><li class="is-active"><a href>Old LSRK Formulations</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Old LSRK Formulations</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/CliMA/ClimaTimeSteppers.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/CliMA/ClimaTimeSteppers.jl/blob/main/docs/src/algorithm_formulations/lsrk.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="Low-storage-Runge–Kutta-methods"><a class="docs-heading-anchor" href="#Low-storage-Runge–Kutta-methods">Low-storage Runge–Kutta methods</a><a id="Low-storage-Runge–Kutta-methods-1"></a><a class="docs-heading-anchor-permalink" href="#Low-storage-Runge–Kutta-methods" title="Permalink"></a></h1><p>LSRK methods are self-starting, with <span>$U^{(1)} = u^n$</span>, and then using stage updates of the form</p><p class="math-container">\[\begin{aligned}
dU^{(i)} &amp;= f(U^{(i)}, t + c_i \Delta t) + A_i dU^{(i-1)}\\
U^{(i+1)} &amp;= U^{(i)} + \Delta t B_i dU^{(i)}
\end{aligned}\]</p><p>where <span>$A_1 = c_1 = 0$</span> (implying <span>$dU^{(1)} = f(u^n, t)$</span>), with the value at the next step being the <span>$N+1$</span>th stage value <span>$u^{n+1} = U^{(N+1)})$</span>.</p><p>This allows the updates to be performed with only two copies of the state vector (so long as <code>f</code> can be evaluated in incrementing form).</p><p>It can be written as an RK scheme with Butcher tableau coefficients defined by the recurrences</p><p class="math-container">\[\begin{aligned}
a_{i,i} &amp;= 0 \\
a_{i,j} &amp;= B_j + A_{j+1} a_{i,j+1}\\
b_N &amp;= B_N \\
b_i &amp;= B_i + A_{i+1} b_{i+1}
\end{aligned}\]</p><p>or equivalently</p><p class="math-container">\[\begin{aligned}
a_{j,j} &amp;= 0 \\
a_{i,j} &amp;= a_{i-1,j} + B_{i-1} \prod_{k=j+1}^{i-1} A_k
\end{aligned}\]</p><p>with <span>$b_j$</span> treated analogously as <span>$a_{N+1,j}$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../rosenbrock/">« Rosenbrock Method</a><a class="docs-footer-nextpage" href="../mrrk/">Old MRRK Formulations »</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="Tuesday 30 July 2024 20:06">Tuesday 30 July 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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