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| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Solvers · StableSpectralElements.jl</title><meta name="title" content="Solvers · StableSpectralElements.jl"/><meta property="og:title" content="Solvers · StableSpectralElements.jl"/><meta property="twitter:title" content="Solvers · StableSpectralElements.jl"/><meta name="description" content="Documentation for StableSpectralElements.jl."/><meta property="og:description" content="Documentation for StableSpectralElements.jl."/><meta property="twitter:description" content="Documentation for StableSpectralElements.jl."/><script data-outdated-warner src="../assets/warner.js"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><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">StableSpectralElements.jl</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="../">Home</a></li><li><span class="tocitem">Modules</span><ul><li><a class="tocitem" href="../ConservationLaws/"><code>ConservationLaws</code></a></li><li><a class="tocitem" href="../SpatialDiscretizations/"><code>SpatialDiscretizations</code></a></li><li class="is-active"><a class="tocitem" href><code>Solvers</code></a><ul class="internal"><li><a class="tocitem" href="#Overview"><span>Overview</span></a></li><li><a class="tocitem" href="#Reference"><span>Reference</span></a></li></ul></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">Modules</a></li><li class="is-active"><a href><code>Solvers</code></a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href><code>Solvers</code></a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/tristanmontoya/StableSpectralElements.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/tristanmontoya/StableSpectralElements.jl/blob/main/docs/src/Solvers.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="Module-Solvers"><a class="docs-heading-anchor" href="#Module-Solvers">Module <code>Solvers</code></a><a id="Module-Solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Module-Solvers" title="Permalink"></a></h1><h2 id="Overview"><a class="docs-heading-anchor" href="#Overview">Overview</a><a id="Overview-1"></a><a class="docs-heading-anchor-permalink" href="#Overview" title="Permalink"></a></h2><p>StableSpectralElements.jl is based on a semi-discrete approach to the numerical solution of partial differential equations, in which the spatial discretization is performed first in order to obtain a system of ordinary differential equations of the form </p><p class="math-container">\[\frac{\mathrm{d} }{\mathrm{d}t}\underline{u}(t) = \underline{R}(\underline{u}(t),t),\]</p><p>where <span>$\underline{u}(t) \in \mathbb{R}^{N_p \cdot N_c \cdot N_e}$</span> is the global solution array containing the <span>$N_p$</span> coefficients for each of the <span>$N_c$</span> solution variables and each of the <span>$N_e$</span> mesh elements. Such systems can be solved using standard time-marching methods from <a href="https://github.com/SciML/OrdinaryDiffEq.jl">OrdinaryDiffEq.jl</a> by computing the semi-discrete residual <span>$\underline{R}(\underline{u}(t),t)$</span> using a Julia function with the following signature:</p><pre><code class="language-julia hljs">semi_discrete_residual(dudt::AbstractArray{Float64,3}, | ||
| u::AbstractArray{Float64, 3}, | ||
| solver::Solver, | ||
| t::Float64)</code></pre><p>The first parameter contains the time derivative to be computed in place, the second parameter is the current solution state, and the fourth parameter is the time <span>$t$</span>. The third parameter, which is of type <code>Solver</code>, contains the all the information defining the spatial discretization as well as preallocated arrays used for temporary storage. The particular algorithm used for computing the semi-discrete residual is then dispatched based on the particular parametric subtype of <code>Solver</code> which is passed into the <code>semi_discrete_residual!</code> function.</p><h2 id="Reference"><a class="docs-heading-anchor" href="#Reference">Reference</a><a id="Reference-1"></a><a class="docs-heading-anchor-permalink" href="#Reference" title="Permalink"></a></h2></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SpatialDiscretizations/">« <code>SpatialDiscretizations</code></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 1.7.0 on <span class="colophon-date" title="Saturday 5 October 2024 22:23">Saturday 5 October 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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