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{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-12T21:03:58","documenter_version":"1.8.0"}}
{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-12T21:07:05","documenter_version":"1.8.0"}}
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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Tutorial · BloodFlowTrixi.jl</title><meta name="title" content="Tutorial · BloodFlowTrixi.jl"/><meta property="og:title" content="Tutorial · BloodFlowTrixi.jl"/><meta property="twitter:title" content="Tutorial · BloodFlowTrixi.jl"/><meta name="description" content="Documentation for BloodFlowTrixi.jl."/><meta property="og:description" content="Documentation for BloodFlowTrixi.jl."/><meta property="twitter:description" content="Documentation for BloodFlowTrixi.jl."/><meta property="og:url" content="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><meta property="twitter:url" content="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><link rel="canonical" href="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><script data-outdated-warner src="../assets/warner.js"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 src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" 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="../">BloodFlowTrixi.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 class="is-active"><a class="tocitem" href>Tutorial</a><ul class="internal"><li><a class="tocitem" href="#Packages"><span>Packages</span></a></li><li><a class="tocitem" href="#Mesh-and-boundary-conditions"><span>Mesh and boundary conditions</span></a></li><li><a class="tocitem" href="#Numerical-flux"><span>Numerical flux</span></a></li><li><a class="tocitem" href="#Basis-functions-and-Shock-Capturing-DG-scheme"><span>Basis functions and Shock Capturing DG scheme</span></a></li><li><a class="tocitem" href="#Semi-discretization"><span>Semi-discretization</span></a></li><li><a class="tocitem" href="#Initial-condition"><span>Initial condition</span></a></li><li><a class="tocitem" href="#Run-the-simulation"><span>Run the simulation</span></a></li><li><a class="tocitem" href="#Plot-the-results"><span>Plot the results</span></a></li><li><a class="tocitem" href="#Plain-code"><span>Plain code</span></a></li><li class="toplevel"><a class="tocitem" href="#Tutorial-for-the-2D-model"><span>Tutorial for the 2D model</span></a></li><li><a class="tocitem" href="#Packages-2"><span>Packages</span></a></li><li><a class="tocitem" href="#Mesh-and-boundary-conditions-2"><span>Mesh and boundary conditions</span></a></li><li><a class="tocitem" href="#Numerical-flux-2"><span>Numerical flux</span></a></li><li><a class="tocitem" href="#Basis-functions-and-Shock-Capturing-DG-scheme-2"><span>Basis functions and Shock Capturing DG scheme</span></a></li><li><a class="tocitem" href="#Semi-discretization-2"><span>Semi-discretization</span></a></li><li><a class="tocitem" href="#Initial-condition-2"><span>Initial condition</span></a></li><li><a class="tocitem" href="#Run-the-simulation-2"><span>Run the simulation</span></a></li><li><a class="tocitem" href="#Plot-the-results-2"><span>Plot the results</span></a></li><li><a class="tocitem" href="#Plain-code-2"><span>Plain code</span></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 class="is-active"><a href>Tutorial</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Tutorial</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/yolhan83/BloodFlowTrixi.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/yolhan83/BloodFlowTrixi.jl/blob/master/docs/src/tuto.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="Tutorial-for-the-1D-model"><a class="docs-heading-anchor" href="#Tutorial-for-the-1D-model">Tutorial for the 1D model</a><a id="Tutorial-for-the-1D-model-1"></a><a class="docs-heading-anchor-permalink" href="#Tutorial-for-the-1D-model" title="Permalink"></a></h1><p>In this section, we describe how to use <strong>BloodFlowTrixi.jl</strong> with <strong>Trixi.jl</strong>. This tutorial will guide you through setting up and running a 1D blood flow simulation, including mesh creation, boundary conditions, numerical fluxes, and visualization of results.</p><h2 id="Packages"><a class="docs-heading-anchor" href="#Packages">Packages</a><a id="Packages-1"></a><a class="docs-heading-anchor-permalink" href="#Packages" title="Permalink"></a></h2><p>Before starting, ensure that the required packages are loaded:</p><pre><code class="language-julia hljs">using Trixi
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Tutorial · BloodFlowTrixi.jl</title><meta name="title" content="Tutorial · BloodFlowTrixi.jl"/><meta property="og:title" content="Tutorial · BloodFlowTrixi.jl"/><meta property="twitter:title" content="Tutorial · BloodFlowTrixi.jl"/><meta name="description" content="Documentation for BloodFlowTrixi.jl."/><meta property="og:description" content="Documentation for BloodFlowTrixi.jl."/><meta property="twitter:description" content="Documentation for BloodFlowTrixi.jl."/><meta property="og:url" content="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><meta property="twitter:url" content="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><link rel="canonical" href="https://yolhan83.github.io/BloodFlowTrixi.jl/tuto/"/><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 src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" 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="../">BloodFlowTrixi.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 class="is-active"><a class="tocitem" href>Tutorial</a><ul class="internal"><li><a class="tocitem" href="#Packages"><span>Packages</span></a></li><li><a class="tocitem" href="#Mesh-and-boundary-conditions"><span>Mesh and boundary conditions</span></a></li><li><a class="tocitem" href="#Numerical-flux"><span>Numerical flux</span></a></li><li><a class="tocitem" href="#Basis-functions-and-Shock-Capturing-DG-scheme"><span>Basis functions and Shock Capturing DG scheme</span></a></li><li><a class="tocitem" href="#Semi-discretization"><span>Semi-discretization</span></a></li><li><a class="tocitem" href="#Initial-condition"><span>Initial condition</span></a></li><li><a class="tocitem" href="#Run-the-simulation"><span>Run the simulation</span></a></li><li><a class="tocitem" href="#Plot-the-results"><span>Plot the results</span></a></li><li><a class="tocitem" href="#Plain-code"><span>Plain code</span></a></li><li class="toplevel"><a class="tocitem" href="#Tutorial-for-the-2D-model"><span>Tutorial for the 2D model</span></a></li><li><a class="tocitem" href="#Packages-2"><span>Packages</span></a></li><li><a class="tocitem" href="#Mesh-and-boundary-conditions-2"><span>Mesh and boundary conditions</span></a></li><li><a class="tocitem" href="#Numerical-flux-2"><span>Numerical flux</span></a></li><li><a class="tocitem" href="#Basis-functions-and-Shock-Capturing-DG-scheme-2"><span>Basis functions and Shock Capturing DG scheme</span></a></li><li><a class="tocitem" href="#Semi-discretization-2"><span>Semi-discretization</span></a></li><li><a class="tocitem" href="#Initial-condition-2"><span>Initial condition</span></a></li><li><a class="tocitem" href="#Run-the-simulation-2"><span>Run the simulation</span></a></li><li><a class="tocitem" href="#Plot-the-results-2"><span>Plot the results</span></a></li><li><a class="tocitem" href="#Plain-code-2"><span>Plain code</span></a></li></ul></li><li><a class="tocitem" href="../math/">Mathematics</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 class="is-active"><a href>Tutorial</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Tutorial</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/yolhan83/BloodFlowTrixi.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/yolhan83/BloodFlowTrixi.jl/blob/master/docs/src/tuto.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="Tutorial-for-the-1D-model"><a class="docs-heading-anchor" href="#Tutorial-for-the-1D-model">Tutorial for the 1D model</a><a id="Tutorial-for-the-1D-model-1"></a><a class="docs-heading-anchor-permalink" href="#Tutorial-for-the-1D-model" title="Permalink"></a></h1><p>In this section, we describe how to use <strong>BloodFlowTrixi.jl</strong> with <strong>Trixi.jl</strong>. This tutorial will guide you through setting up and running a 1D blood flow simulation, including mesh creation, boundary conditions, numerical fluxes, and visualization of results.</p><h2 id="Packages"><a class="docs-heading-anchor" href="#Packages">Packages</a><a id="Packages-1"></a><a class="docs-heading-anchor-permalink" href="#Packages" title="Permalink"></a></h2><p>Before starting, ensure that the required packages are loaded:</p><pre><code class="language-julia hljs">using Trixi
using BloodFlowTrixi
using OrdinaryDiffEq
using Plots</code></pre><p>First, we need to choose the equation that describes the blood flow dynamics:</p><pre><code class="language-julia hljs">eq = BloodFlowEquations1D(; h=0.1)</code></pre><p>Here, <code>h</code> represents a parameter related to the initial condition or model scaling.</p><h2 id="Mesh-and-boundary-conditions"><a class="docs-heading-anchor" href="#Mesh-and-boundary-conditions">Mesh and boundary conditions</a><a id="Mesh-and-boundary-conditions-1"></a><a class="docs-heading-anchor-permalink" href="#Mesh-and-boundary-conditions" title="Permalink"></a></h2><p>We begin by defining a one-dimensional Tree mesh, which discretizes the spatial domain:</p><pre><code class="language-julia hljs">mesh = TreeMesh(0.0, 40.0, initial_refinement_level=6, n_cells_max=10^4, periodicity=false)</code></pre><p>This generates a non-periodic mesh for the interval <span>$[0, 40]$</span>, with <span>$2^{initial-refinement-level+1}-1$</span> cells. The parameter <code>initial_refinement_level</code> controls the initial number of cells, while <code>n_cells_max</code> specifies the maximum number of cells allowed during mesh refinement.</p><p>In <strong>Trixi.jl</strong>, the Tree mesh has two labeled boundaries: ***x<em>neg*** (left boundary) and ***x</em>pos*** (right boundary). These labels are used to apply boundary conditions:</p><pre><code class="language-julia hljs">bc = (
Expand Down Expand Up @@ -148,4 +148,4 @@
plt3 = Plots.plot(pd[&quot;ws&quot;],aspect_ratio=0.2)
plt4 = Plots.plot(pd[&quot;P&quot;],aspect_ratio=0.2)
plot(plt1,plt2,plt3,plt4,layout=(2,2))
end</code></pre><p><img src="../graph2d.gif" alt="Alt Text"/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</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 1.8.0 on <span class="colophon-date" title="Sunday 12 January 2025 21:03">Sunday 12 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
end</code></pre><p><img src="../graph2d.gif" alt="Alt Text"/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../math/">Mathematics »</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 1.8.0 on <span class="colophon-date" title="Sunday 12 January 2025 21:07">Sunday 12 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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