deploy: b3d5a19

Author: hweifluids

.doctrees/3_numerical.doctree

@@ -126,9 +126,22 @@
 \[y_{n+1} = y_n + h \sum_{i=1}^s b_i\,k_i.\]</div>
 <p><strong>Second-Order Runge-Kutta (RK2, Heun’s)</strong></p>
 <p>Heun’s method attains second-order accuracy by combining predictor and corrector slopes:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{aligned}
+k_1 &amp;= \sigma\,\mathbf{u}(\mathbf{x}_n,t_n),\\
+\mathbf{x}^* &amp;= \mathbf{x}_n + \Delta t\,k_1,\\
+k_2 &amp;= \sigma\,\mathbf{u}(\mathbf{x}^*,t_n + \Delta t),\\
+\mathbf{x}_{n+1} &amp;= \mathbf{x}_n + \tfrac{\Delta t}{2}\,(k_1 + k_2).
+\end{aligned}\end{split}\]</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>The classical RK4 method achieves fourth-order accuracy via four slope evaluations at intermediate points:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
+k_2 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{2}k_1,\;t_n + \tfrac{\Delta t}{2}\bigr),\\
+k_3 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{2}k_2,\;t_n + \tfrac{\Delta t}{2}\bigr),\\
+k_4 = \mathbf{u}(\mathbf{x}_n + \Delta t\,k_3,\;t_n + \Delta t),\\
+\mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\Delta t}{6}\,(k_1 + 2k_2 + 2k_3 + k_4).\end{split}\]</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 <code class="docutils literal notranslate"><span class="pre">ERK6(7)</span></code> uses non-uniform weights to attain global <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> accuracy, originating from <a class="reference internal" href="9_references.html#butcher1964" id="id1"><span>[Butcher1964]</span></a>.
@@ -220,7 +233,7 @@
 </tr>
 </tbody>
 </table>
-<p>In our computation, the up symbol side is applied, in other words, <code class="docutils literal notranslate"><span class="pre">±</span></code> represents <code class="docutils literal notranslate"><span class="pre">+</span></code>, taking <span class="math notranslate nohighlight">\(\lambda=+\sqrt{5}\)</span>. With 15 digis are kept, the Butcher table used by the author is shown in the following table.</p>
+<p>In our computation, the up symbol side is applied, in other words, <code class="docutils literal notranslate"><span class="pre">±</span></code> represents <code class="docutils literal notranslate"><span class="pre">+</span></code>, taking <span class="math notranslate nohighlight">\(\lambda=+\sqrt{5}\)</span>. With 15 digis are kept, the explicit Butcher table for <code class="docutils literal notranslate"><span class="pre">RK6</span></code> used by the author is shown in the following table.</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>
@@ -297,7 +310,7 @@
 <td><p>3.61803398874989</p></td>
 <td><p>0</p></td>
 </tr>
-<tr class="row-odd"><td><p>b_i</p></td>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(b_i\)</span></p></td>
 <td><p>0.0833333333333333</p></td>
 <td><p>0</p></td>
 <td><p>0</p></td>

.doctrees/environment.pickle

@@ -73,12 +73,13 @@ and then form the new approximation by
 Heun's method attains second-order accuracy by combining predictor and corrector slopes:
 
 .. math::
-   :align: left
 
-   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).
+   \begin{aligned}
+   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{aligned}
 
 This scheme yields a global error of order :math:`O(\Delta t^2)` with two velocity evaluations per step.
 
@@ -87,7 +88,6 @@ This scheme yields a global error of order :math:`O(\Delta t^2)` with two veloci
 The classical RK4 method achieves fourth-order accuracy via four slope evaluations at intermediate points:
 
 .. math::
-   :align: left
 
    k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
    k_2 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{2}k_1,\;t_n + \tfrac{\Delta t}{2}\bigr),\\
@@ -122,7 +122,7 @@ As for the coefficients for ``RK6`` are more complex to write into equations, th
 | :math:`b_i`                 | :math:`1/12`                  | :math:`0`                  | :math:`0`                       | :math:`0`                    | :math:`5/12`                | :math:`5/12`             | :math:`1/12`            |
 +-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
 
-In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`. With 15 digis are kept, the Butcher table used by the author is shown in the following table.
+In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`. With 15 digis are kept, the explicit Butcher table for ``RK6`` used by the author is shown in the following table.
 
 +-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
 |  :math:`c_i`                  |  :math:`a_{i1}`               |  :math:`a_{i2}`               |  :math:`a_{i3}`               |  :math:`a_{i4}`               |  :math:`a_{i5}`               |  :math:`a_{i6}`               |  :math:`a_{i7}`               |
@@ -141,7 +141,7 @@ In our computation, the up symbol side is applied, in other words, ``±`` repres
 +-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
 | 1                             | 0.166666666666667             | 0                             | 0.0751416197912285            | -3.38770632020821             | 0.52786404500042              | 3.61803398874989              | 0                             |
 +-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
-| b_i                           | 0.0833333333333333            | 0                             | 0                             | 0                             | 0.416666666666667             | 0.416666666666667             | 0.0833333333333333            |
+| :math:`b_i`                   | 0.0833333333333333            | 0                             | 0                             | 0                             | 0.416666666666667             | 0.416666666666667             | 0.0833333333333333            |
 +-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+