deploy: 9c60813

Author: hweifluids

.doctrees/3_numerical.doctree

@@ -105,36 +105,36 @@
 </section>
 <section id="time-integration">
 <span id="marching"></span><h3>Time Integration<a class="headerlink" href="#time-integration" title="Link to this heading"></a></h3>
-<p>Consider the initial‐value problem for passive tracer advection in a continuous velocity field</p>
+<p>Consider the initial-value problem for passive tracer advection in a continuous velocity field</p>
 <div class="math notranslate nohighlight">
 \[\frac{d\mathbf{x}}{dt} = \sigma\,\mathbf{u}(\mathbf{x},t)\,,
 \mathbf{x}(t_n)=\mathbf{x}_n\,,\]</div>
 <p>where <span class="math notranslate nohighlight">\(\sigma = \pm1\)</span> selects forward or backward integration.</p>
 <p><strong>Explicit Euler Method</strong></p>
-<p>The first‐order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:</p>
+<p>The first-order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}\mathbf{u}_n = \mathbf{u}(\mathbf{x}_n,t_n),\\
 \mathbf{x}_{n+1} = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}_n.\end{split}\]</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>Heun’s method attains second‐order accuracy by combining predictor and corrector slopes:</p>
+<p><strong>Second-Order Runge-Kutta (Heun’s Method)</strong></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>The classical RK4 method achieves fourth‐order accuracy via four slope evaluations at intermediate points:</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 six‐stage scheme uses non‐uniform weights to attain sixth‐order accuracy:</p>
+<p><strong>Sixth-Order Runge-Kutta (RK6)</strong></p>
+<p>The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}k_1 = \mathbf{u}(\mathbf{x}_n,t_n),\\
 k_2 = \mathbf{u}\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{3}k_1,\;t_n + \tfrac{\Delta t}{3}\bigr),\\
@@ -144,7 +144,6 @@
 k_6 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4),\;t_n + \Delta t\bigr),\\
 \mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).\end{split}\]</div>
 <p>This scheme incurs a global error of order <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> with six velocity evaluations.</p>
-<p>All methods assume a continuous velocity interpolation (e.g., tricubic) to supply <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> at arbitrary particle positions and times.</p>
 </section>
 </section>
 <section id="ftle-computation">

.doctrees/environment.pickle

@@ -25,7 +25,7 @@ Wall Treatment
 Time Integration
 ~~~~~~~~~~~~~~~~~~
 
-Consider the initial‐value problem for passive tracer advection in a continuous velocity field
+Consider the initial-value problem for passive tracer advection in a continuous velocity field
 
 .. math::
 
@@ -36,7 +36,7 @@ where :math:`\sigma = \pm1` selects forward or backward integration.
 
 **Explicit Euler Method**
 
-The first‐order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:
+The first-order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:
 
 .. math::
 
@@ -45,9 +45,9 @@ The first‐order explicit Euler scheme advances the position by sampling the ve
 
 This method incurs a global error of order :math:`O(\Delta t)` and requires only one velocity evaluation per step.
 
-**Second‐Order Runge–Kutta (Heun’s Method)**
+**Second-Order Runge-Kutta (Heun's Method)**
 
-Heun’s method attains second‐order accuracy by combining predictor and corrector slopes:
+Heun's method attains second-order accuracy by combining predictor and corrector slopes:
 
 .. math::
 
@@ -58,9 +58,9 @@ Heun’s method attains second‐order accuracy by combining predictor and corre
 
 This scheme yields a global error of order :math:`O(\Delta t^2)` with two velocity evaluations per step.
 
-**Classical Fourth‐Order Runge–Kutta (RK4)**
+**Classical Fourth-Order Runge-Kutta (RK4)**
 
-The classical RK4 method achieves fourth‐order accuracy via four slope evaluations at intermediate points:
+The classical RK4 method achieves fourth-order accuracy via four slope evaluations at intermediate points:
 
 .. math::
 
@@ -72,9 +72,9 @@ The classical RK4 method achieves fourth‐order accuracy via four slope evaluat
 
 This yields a global error of order :math:`O(\Delta t^4)` with four velocity evaluations per step.
 
-**Sixth‐Order Runge–Kutta (RK6)**
+**Sixth-Order Runge-Kutta (RK6)**
 
-The six‐stage scheme uses non‐uniform weights to attain sixth‐order accuracy:
+The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:
 
 .. math::
 
@@ -88,8 +88,6 @@ The six‐stage scheme uses non‐uniform weights to attain sixth‐order accura
 
 This scheme incurs a global error of order :math:`O(\Delta t^6)` with six velocity evaluations.  
 
-All methods assume a continuous velocity interpolation (e.g., tricubic) to supply :math:`\mathbf{u}` at arbitrary particle positions and times.
-
 .. _ftlefinal:
 
 FTLE Computation