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‎.doctrees/3_numerical.doctree‎

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‎.doctrees/9_references.doctree‎

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‎3_numerical.html‎

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\mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\Delta t}{6}\,(k_1 + 2k_2 + 2k_3 + k_4).\end{split}\]</div>
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<p>This yields a global error of order <span class="math notranslate nohighlight">\(O(\Delta t^4)\)</span> with four velocity evaluations per step.</p>
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<p><strong>Sixth-Order Runge-Kutta (RK6)</strong></p>
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<p>The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:</p>
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<p>The seven-stage scheme uses non-uniform weights to attain global <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> accuracy. This method originates from</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
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k_2 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{3}k_1,\;t_n + \tfrac{\Delta t}{3}\bigr),\\
@@ -143,70 +143,72 @@
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k_5 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(\tfrac{1}{2}k_1 - \tfrac{3}{2}k_3 + 2k_4),\;t_n + \tfrac{2\Delta t}{3}\bigr),\\
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k_6 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4),\;t_n + \Delta t\bigr),\\
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\mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).\end{split}\]</div>
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<p>This scheme incurs a global error of order <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> with six velocity evaluations.</p>
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<p>This scheme incurs a global error of order with six velocity evaluations.</p>
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<table class="docutils align-default">
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<thead>
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<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">output_dir</span></code></p></th>
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<th class="head"><p>Path to directory for saving results</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">&lt;string&gt;</span></code></p></th>
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<th class="head"></th>
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</tr>
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<tr class="row-even"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">auto_wall</span></code></p></th>
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<th class="head"><p>Flag for automatic wall detection</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">0</span></code></p></th>
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<th class="head"><p>Manually indicate computation bounds.</p></th>
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</tr>
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<tr class="row-odd"><th class="head"></th>
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<th class="head"></th>
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<th class="head"><p>(<code class="docutils literal notranslate"><span class="pre">1</span></code>)</p></th>
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<th class="head"><p>Auto detect bounds from input mesh.</p></th>
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</tr>
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<tr class="row-even"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">wall_xrange</span></code></p></th>
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<th class="head"><p>X-direction wall bounds [xmin, xmax]</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">[&lt;float&gt;,</span> <span class="pre">&lt;float&gt;]</span></code></p></th>
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<th class="head"><p>Dimensionalized coordinates bounds.</p></th>
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</tr>
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<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">wall_yrange</span></code></p></th>
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<th class="head"><p>Y-direction wall bounds [ymin, ymax]</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">[&lt;float&gt;,</span> <span class="pre">&lt;float&gt;]</span></code></p></th>
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<th class="head"></th>
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<tbody>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">wall_zrange</span></code></p></th>
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<th class="head"><p>Z-direction wall bounds [zmin, zmax]</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">[&lt;float&gt;,</span> <span class="pre">&lt;float&gt;]</span></code></p></th>
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<th class="head"></th>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">x</span></code></p></th>
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<th class="head"><p>Number of Lagrangian seed points in X direction</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">&lt;int&gt;</span></code></p></th>
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<th class="head"></th>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(\mp\sqrt{5}/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm2\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">y</span></code></p></th>
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<th class="head"><p>Number of Lagrangian seed points in Y direction</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">&lt;int&gt;</span></code></p></th>
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<th class="head"></th>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-15\pm7\sqrt{5})/20\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-1\pm\sqrt{5})/4\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((15\mp7\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">z</span></code></p></th>
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<th class="head"><p>Number of Lagrangian seed points in Z direction</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">&lt;int&gt;</span></code></p></th>
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<th class="head"></th>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/60\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((15\pm7\sqrt{5})/60\)</span></p></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">T_range</span></code></p></th>
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<th class="head"><p>Time range [<code class="docutils literal notranslate"><span class="pre">T_start</span></code>, <code class="docutils literal notranslate"><span class="pre">T_end</span></code>] in frames</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">[&lt;float&gt;,</span> <span class="pre">&lt;float&gt;]</span></code></p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">T_end</span></code> no larger than <code class="docutils literal notranslate"><span class="pre">N_frames</span></code>.</p></th>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/60\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((9\mp5\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-5\pm3\sqrt{5})/10\)</span></p></td>
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<td></td>
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</tr>
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<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">dt</span></code></p></th>
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<th class="head"><p>Time step for particle advection in frames</p></th>
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<th class="head"><p><code class="docutils literal notranslate"><span class="pre">&lt;float&gt;</span></code></p></th>
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<th class="head"></th>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-55\pm25\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-25\mp7\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(5\mp2\sqrt{5}\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/2\)</span></p></td>
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</tr>
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</thead>
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<tbody>
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<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">physical_dt</span></code></p></td>
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<td><p>Physical time step between two input frames</p></td>
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<td><p><code class="docutils literal notranslate"><span class="pre">&lt;float&gt;</span></code></p></td>
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<td><p>Perform physical advection.</p></td>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(1/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(5/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(5/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/12\)</span></p></td>
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</tbody>
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</table>

‎9_references.html‎

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<span class="label"><span class="fn-bracket">[</span>Lekien2005<span class="fn-bracket">]</span></span>
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<p>Lekien, F., Marsden, J. “Tricubic interpolation in three dimensions,” <em>International Journal for Numerical Methods in Engineering</em> <strong>63</strong> (2005).</p>
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</div>
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<div class="citation" id="butcher" role="doc-biblioentry">
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<span class="label"><span class="fn-bracket">[</span>Butcher<span class="fn-bracket">]</span></span>
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<p>Butcher J. C. “On Runge-Kutta processes of high order.” <em>Journal of the Australian Mathematical Society</em> <strong>4(2)</strong> (1964). doi:10.1017/S1446788700023387</p>
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</div>
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</div>
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<section id="papers-who-uses-py3dftle">
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<h2>Papers Who Uses <code class="docutils literal notranslate"><span class="pre">Py3DFTLE</span></code><a class="headerlink" href="#papers-who-uses-py3dftle" title="Link to this heading"></a></h2>

‎_sources/3_numerical.rst.txt‎

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**Sixth-Order Runge-Kutta (RK6)**
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The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:
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The seven-stage scheme uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from
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.. math::
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k_6 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4),\;t_n + \Delta t\bigr),\\
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\mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).
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This scheme incurs a global error of order :math:`O(\Delta t^6)` with six velocity evaluations.
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``output_dir`` | Path to directory for saving results | ``<string>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``auto_wall`` | Flag for automatic wall detection | ``0`` | Manually indicate computation bounds. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| | | (``1``) | Auto detect bounds from input mesh. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_xrange`` | X-direction wall bounds [xmin, xmax] | ``[<float>, <float>]`` | Dimensionalized coordinates bounds. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_yrange`` | Y-direction wall bounds [ymin, ymax] | ``[<float>, <float>]`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_zrange`` | Z-direction wall bounds [zmin, zmax] | ``[<float>, <float>]`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``x`` | Number of Lagrangian seed points in X direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``y`` | Number of Lagrangian seed points in Y direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``z`` | Number of Lagrangian seed points in Z direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``T_range`` | Time range [``T_start``, ``T_end``] in frames | ``[<float>, <float>]`` | ``T_end`` no larger than ``N_frames``. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``dt`` | Time step for particle advection in frames | ``<float>`` | |
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+===================+======================================================+=========================+=========================================================================================================+
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| ``physical_dt`` | Physical time step between two input frames | ``<float>`` | Perform physical advection. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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This scheme incurs a global error of order with six velocity evaluations.
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`0` | | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/10` | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`\mp\sqrt{5}/10` | :math:`(5\pm2\sqrt{5})/10` | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(-15\pm7\sqrt{5})/20` | :math:`(-1\pm\sqrt{5})/4` | :math:`(15\mp7\sqrt{5})/10` | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/60` | :math:`0` | :math:`1/6` | :math:`(15\pm7\sqrt{5})/60` | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\pm\sqrt{5})/60` | :math:`0` | :math:`(9\mp5\sqrt{5})/12` | :math:`1/6` | :math:`(-5\pm3\sqrt{5})/10` | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`1` | :math:`1/6` | :math:`0` | :math:`(-55\pm25\sqrt{5})/12` | :math:`(-25\mp7\sqrt{5})/12` | :math:`5\mp2\sqrt{5}` | :math:`(5\pm\sqrt{5})/2` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`1/12` | :math:`0` | :math:`0` | :math:`0` | :math:`5/12` | :math:`5/12` | :math:`1/12` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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.. _ftlefinal:

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