deploy: cf856f4

Author: hweifluids

.doctrees/3_numerical.doctree

@@ -116,15 +116,26 @@
 \[\begin{split}\mathbf{u}_n = \mathbf{u}(\mathbf{x}_n,t_n),\\
 \mathbf{x}_{n+1} = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}_n.\end{split}\]</div>
 <p>This method incurs a global error of order <span class="math notranslate nohighlight">\(O(\Delta t)\)</span> and requires only one velocity evaluation per step.</p>
-<p><strong>Second-Order Runge-Kutta (Heun’s Method)</strong></p>
+<p><strong>Runge-Kutta Method</strong></p>
+<p>In an explicit <span class="math notranslate nohighlight">\(s\)</span>-stage Runge–Kutta method for the initial-value problem:</p>
+<div class="math notranslate nohighlight">
+\[y' = f(t,y), \quad y(t_n) = y_n\]</div>
+<p>one advances the solution by a step <span class="math notranslate nohighlight">\(h\)</span> as follows. First compute the intermediate slopes</p>
+<div class="math notranslate nohighlight">
+\[k_i = f\Bigl(t_n + c_i\,h,\;y_n + h \sum_{j=1}^{i-1} a_{ij}\,k_j\Bigr),
+\quad i = 1,2,\dots,s\]</div>
+<p>and then form the new approximation by</p>
+<div class="math notranslate nohighlight">
+\[y_{n+1} = y_n + h \sum_{i=1}^s b_i\,k_i.\]</div>
+<p><em>Second-Order Runge-Kutta (RK2, Heun’s)</em></p>
 <p>Heun’s method attains second-order accuracy by combining predictor and corrector slopes:</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}k_1 = \sigma\,\mathbf{u}(\mathbf{x}_n,t_n),\\
 \mathbf{x}^* = \mathbf{x}_n + \Delta t\,k_1,\\
 k_2 = \sigma\,\mathbf{u}(\mathbf{x}^*,t_n + \Delta t),\\
 \mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\Delta t}{2}\,(k_1 + k_2).\end{split}\]</div>
 <p>This scheme yields a global error of order <span class="math notranslate nohighlight">\(O(\Delta t^2)\)</span> with two velocity evaluations per step.</p>
-<p><strong>Classical Fourth-Order Runge-Kutta (RK4)</strong></p>
+<p><em>Classical Fourth-Order Runge-Kutta (RK4)</em></p>
 <p>The classical RK4 method achieves fourth-order accuracy via four slope evaluations at intermediate points:</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
@@ -133,83 +144,86 @@
 k_4 = \mathbf{u}(\mathbf{x}_n + \Delta t\,k_3,\;t_n + \Delta t),\\
 \mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\Delta t}{6}\,(k_1 + 2k_2 + 2k_3 + k_4).\end{split}\]</div>
 <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>
-<p><strong>Sixth-Order Runge-Kutta (RK6)</strong></p>
-<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 <a class="reference internal" href="9_references.html#butcher" id="id1"><span>[Butcher]</span></a>, as well as the coefficients (Butcher table) listed in the basically-tailest table of the reference.</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
-k_2 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{3}k_1,\;t_n + \tfrac{\Delta t}{3}\bigr),\\
-k_3 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(\tfrac{1}{6}k_1 + \tfrac{1}{6}k_2),\;t_n + \tfrac{\Delta t}{3}\bigr),\\
-k_4 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(\tfrac{1}{8}k_1 + \tfrac{3}{8}k_3),\;t_n + \tfrac{\Delta t}{2}\bigr),\\
-k_5 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(\tfrac{1}{2}k_1 - \tfrac{3}{2}k_3 + 2k_4),\;t_n + \tfrac{2\Delta t}{3}\bigr),\\
-k_6 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4),\;t_n + \Delta t\bigr),\\
-\mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).\end{split}\]</div>
-<p>This scheme incurs a global error of order  with six velocity evaluations.</p>
+<p><em>Sixth-Order Runge-Kutta (RK6)</em></p>
+<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 <a class="reference internal" href="9_references.html#butcher" id="id1"><span>[Butcher]</span></a>.
+As for the coefficients for <code class="docutils literal notranslate"><span class="pre">RK6</span></code> are more complex to write into equations, the Butcher table is given as follows.</p>
 <table class="docutils align-default">
+<thead>
+<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(c_i\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i1}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i2}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i3}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i4}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i5}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i6}\)</span></p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(a_{i7}\)</span></p></th>
+</tr>
+</thead>
 <tbody>
-<tr class="row-odd"><td></td>
-<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
 </tr>
-<tr class="row-even"><td></td>
-<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
+<td></td>
 </tr>
-<tr class="row-odd"><td></td>
-<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(\mp\sqrt{5}/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((5\pm2\sqrt{5})/10\)</span></p></td>
 <td></td>
 <td></td>
 <td></td>
 <td></td>
+<td></td>
 </tr>
-<tr class="row-even"><td></td>
-<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((-15\pm7\sqrt{5})/20\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((-1\pm\sqrt{5})/4\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((15\mp7\sqrt{5})/10\)</span></p></td>
 <td></td>
 <td></td>
 <td></td>
+<td></td>
 </tr>
-<tr class="row-odd"><td></td>
-<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/60\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((15\pm7\sqrt{5})/60\)</span></p></td>
 <td></td>
 <td></td>
+<td></td>
 </tr>
-<tr class="row-even"><td></td>
-<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/60\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((9\mp5\sqrt{5})/12\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((-5\pm3\sqrt{5})/10\)</span></p></td>
 <td></td>
+<td></td>
 </tr>
-<tr class="row-odd"><td></td>
-<td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((-55\pm25\sqrt{5})/12\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((-25\mp7\sqrt{5})/12\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(5\mp2\sqrt{5}\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/2\)</span></p></td>
+<td></td>
 </tr>
-<tr class="row-even"><td></td>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(b_i\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(1/12\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
 <td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>