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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Fractional step method · Oceananigans.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link rel="canonical" href="https://clima.github.io/OceananigansDocumentation/stable/appendix/fractional_step/"/><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/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link 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class="tocitem" href="../staggered_grid/">Staggered grid</a></li><li class="is-active"><a class="tocitem" href>Fractional step method</a></li><li><a class="tocitem" href="../convergence_tests/">Convergence tests</a></li><li><a class="tocitem" href="../benchmarks/">Performance benchmarks</a></li><li><a class="tocitem" href="../library/">Library</a></li><li><a class="tocitem" href="../function_index/">Function index</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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Appendix</a></li><li class="is-active"><a href>Fractional step method</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Fractional step method</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/CliMA/Oceananigans.jl/blob/main/docs/src/appendix/fractional_step.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="fractional_step_method"><a class="docs-heading-anchor" href="#fractional_step_method">Fractional step method</a><a id="fractional_step_method-1"></a><a class="docs-heading-anchor-permalink" href="#fractional_step_method" title="Permalink"></a></h1><p>In some models (e.g., <code>NonhydrostaticModel</code> or <code>HydrostaticFreeSurfaceModel</code>) solving the momentum coupled with the continuity equation can be cumbersome so instead we employ a fractional step method. To approximate the solution of the coupled system we first solve an approximation to the discretized momentum equation for an intermediate velocity field <span>$\boldsymbol{v}^\star$</span> without worrying about satisfying the incompressibility constraint. We then project <span>$\boldsymbol{v}^\star$</span> onto the space of divergence-free velocity fields to obtain a value for <span>$\boldsymbol{v}^{n+1}$</span> that satisfies continuity.</p><p>For example, for the <code>NonhydrostaticModel</code>, if we ignore the background velocity fields and the surface waves, we thus discretize the momentum equation as</p><p class="math-container">\[ \frac{\boldsymbol{v}^\star - \boldsymbol{v}^n}{\Delta t}
= - \left[ \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v} \right]^{n+\frac{1}{2}}
- \boldsymbol{f} \times \boldsymbol{v}^{n+\frac{1}{2}}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left ( \nu \boldsymbol{\nabla} \boldsymbol{v}^{n+\frac{1}{2}} \right )
+ \boldsymbol{F}_{\boldsymbol{v}}^{n+\frac{1}{2}} \, ,\]</p><p>where the superscript <span>$n + \frac{1}{2}$</span> indicates that these terms are evaluated at time step <span>$n + \frac{1}{2}$</span>, which we compute explicitly (see <a href="../../numerical_implementation/time_stepping/#time_stepping">Time-stepping section</a>).</p><p>The projection is then performed</p><p class="math-container">\[ \boldsymbol{v}^{n+1} = \boldsymbol{v}^\star - \Delta t \, \boldsymbol{\nabla} p^{n+1} \, ,\]</p><p>to obtain a divergence-free velocity field <span>$\boldsymbol{v}^{n+1}$</span>. Here the projection is performed by solving an elliptic problem for the pressure <span>$p^{n+1}$</span> with the boundary condition</p><p class="math-container">\[ \boldsymbol{\hat{n}} \boldsymbol{\cdot} \boldsymbol{\nabla} p^{n+1} |_{\partial\Omega} = 0 \, .\]</p><p><a href="../../references/#Orszag86">Orszag <em>et al.</em> (1986)</a> and <a href="../../references/#Brown01">Brown <em>et al.</em> (2001)</a> raise an important issue regarding these fractional step methods, which is that &quot;<em>while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the <span>$L_\infty$</span>-norm.</em>&quot; The numerical boundary conditions must be carefully accounted for to ensure the second-order accuracy promised by the fractional step methods.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../staggered_grid/">« Staggered grid</a><a class="docs-footer-nextpage" href="../convergence_tests/">Convergence tests »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 27 September 2023 19:55">Wednesday 27 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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