deploy: e822044

Author: hweifluids

.doctrees/3_numerical.doctree

@@ -120,13 +120,13 @@
 First, compute the intermediate stage vectors:</p>
 <div class="math notranslate nohighlight">
 \[\mathbf{k}_i =
-\mathbf{u} \Bigl(   \mathbf{x}_n + \Delta t \sum_{j=1}^{i-1} a_{ij}\,\mathbf{k}_j,\,
+\mathbf{u} \Bigl(   \mathbf{x}_n + \sigma\,\Delta t \sum_{j=1}^{i-1} a_{ij}\,\mathbf{k}_j,\,
 t_n + c_i \Delta t \Bigr),
 \qquad i = 1, 2, \dots, s,\]</div>
 <p>and then update the solution:</p>
 <div class="math notranslate nohighlight">
 \[\mathbf{x}_{n+1} =
-\mathbf{x}_n + \Delta t \sum_{i=1}^s b_i\,\mathbf{k}_i.\]</div>
+\mathbf{x}_n + \sigma\,\Delta t \sum_{i=1}^s b_i\,\mathbf{k}_i.\]</div>
 <p>Here, the boldface stage variables <span class="math notranslate nohighlight">\(\mathbf{k}_i\)</span> represent intermediate slope estimates.</p>
 <p><strong>Second-Order Runge-Kutta (RK2, Heun’s)</strong></p>
 <p>Heun’s <code class="docutils literal notranslate"><span class="pre">RK2</span></code> method attains second-order accuracy by combining predictor and corrector slopes:</p>

.doctrees/environment.pickle

@@ -51,7 +51,7 @@ First, compute the intermediate stage vectors:
 .. math::
 
    \mathbf{k}_i = 
-   \mathbf{u} \Bigl(   \mathbf{x}_n + \Delta t \sum_{j=1}^{i-1} a_{ij}\,\mathbf{k}_j,\,
+   \mathbf{u} \Bigl(   \mathbf{x}_n + \sigma\,\Delta t \sum_{j=1}^{i-1} a_{ij}\,\mathbf{k}_j,\,
    t_n + c_i \Delta t \Bigr),
    \qquad i = 1, 2, \dots, s,
 
@@ -60,7 +60,7 @@ and then update the solution:
 .. math::
 
    \mathbf{x}_{n+1} =
-   \mathbf{x}_n + \Delta t \sum_{i=1}^s b_i\,\mathbf{k}_i.
+   \mathbf{x}_n + \sigma\,\Delta t \sum_{i=1}^s b_i\,\mathbf{k}_i.
 
 Here, the boldface stage variables :math:`\mathbf{k}_i` represent intermediate slope estimates.