time integration

Author: hweifluids

source/3_numerical.rst

@@ -11,6 +11,7 @@ Lagrangian Advection
 
 
 .. _intp:
+
 Velocity Interpolation
 ~~~~~~~~~~~~~~~~~~~~~~~
 
@@ -21,20 +22,110 @@ Wall Treatment
 
 .. _marching:
 
-Particle Marching
+Time Integration
 ~~~~~~~~~~~~~~~~~~
 
+Consider the initial-value problem for particle advection in a velocity field
 
+.. math::
 
-.. _ftlefinal:
+   \frac{d\mathbf{x}}{dt} = \sigma\,\mathbf{u}(\mathbf{x},t),
+   \quad \mathbf{x}(t_n) = \mathbf{x}_n,
 
-FTLE Computation
--------------------
+where :math:`\sigma = \pm 1` indicates forward/backward advection.
+
+**Explicit Euler Method**
+
+Given :math:`\mathbf{x}_n` at time :math:`t_n`, the first-order (Euler) update is
+
+.. math::
+
+   \mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\,f(\mathbf{x}_n,t_n)
+                     = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}(\mathbf{x}_n,t_n)
+
+Implementation steps:
+
+  1. Evaluate velocity:  
+     .. math::  
+        \mathbf{u}_n = \mathbf{u}(\mathbf{x}_n, t_n)  
+  2. Advance position:  
+     .. math::  
+        \mathbf{x}_{n+1} = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}_n  
+
+This method is simple but incurs :math:`O(\Delta t)` local truncation error.
+
+**Second-Order Runge–Kutta (Heun’s Method)**
+
+A two-stage explicit scheme with :math:`O(\Delta t^2)` accuracy:
+
+.. math::
+
+   k_1 &= f(\mathbf{x}_n, t_n),\\
+   \mathbf{x}^* &= \mathbf{x}_n + \Delta t\,k_1,\\
+   k_2 &= f(\mathbf{x}^*, t_n + \Delta t),\\
+   \mathbf{x}_{n+1} &= \mathbf{x}_n + \tfrac{\Delta t}{2}\,(k_1 + k_2).
+
+Implementation steps:
 
+  1. Compute :math:`k_1 = \sigma\,\mathbf{u}(\mathbf{x}_n, t_n)`.  
+  2. Predict step:  
+     .. math::  
+        \mathbf{x}^* = \mathbf{x}_n + \sigma\,\Delta t\,k_1  
+  3. Compute :math:`k_2 = \sigma\,\mathbf{u}(\mathbf{x}^*, t_n + \Delta t)`.  
+  4. Update:  
+     .. math::  
+        \mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\sigma\,\Delta t}{2}\,(k_1 + k_2)
 
+**Classical Fourth-Order Runge–Kutta (RK4)**
 
+A four-stage scheme with :math:`O(\Delta t^4)` accuracy:
 
+.. math::
 
+   k_1 &= f(\mathbf{x}_n, t_n),\\
+   k_2 &= f\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{2}k_1,\;t_n + \tfrac{\Delta t}{2}\bigr),\\
+   k_3 &= f\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{2}k_2,\;t_n + \tfrac{\Delta t}{2}\bigr),\\
+   k_4 &= f(\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).
+
+Implementation steps:
+
+  1. Compute :math:`k_1` at :math:`\mathbf{x}_n`.  
+  2. Compute :math:`k_2` at :math:`\mathbf{x}_n + \tfrac{\Delta t}{2}k_1`.  
+  3. Compute :math:`k_3` at :math:`\mathbf{x}_n + \tfrac{\Delta t}{2}k_2`.  
+  4. Compute :math:`k_4` at :math:`\mathbf{x}_n + \Delta t\,k_3`.  
+  5. Combine:  
+     .. math::  
+        \mathbf{x}_{n+1} = \mathbf{x}_n + \tfrac{\Delta t}{6}\,(k_1 + 2k_2 + 2k_3 + k_4)
+
+**Sixth-Order Runge–Kutta (RK6)**
+
+A six-stage explicit method with :math:`O(\Delta t^6)` accuracy. Define:
+
+.. math::
+
+   k_1 &= f(\mathbf{x}_n, t_n),\\
+   k_2 &= f\!\bigl(\mathbf{x}_n + \tfrac{\Delta t}{3}k_1,\;t_n + \tfrac{\Delta t}{3}\bigr),\\
+   k_3 &= f\!\Bigl(\mathbf{x}_n + \Delta t\bigl(\tfrac{1}{6}k_1 + \tfrac{1}{6}k_2\bigr),\;t_n + \tfrac{\Delta t}{3}\Bigr),\\
+   k_4 &= f\!\Bigl(\mathbf{x}_n + \Delta t\bigl(\tfrac{1}{8}k_1 + \tfrac{3}{8}k_3\bigr),\;t_n + \tfrac{\Delta t}{2}\Bigr),\\
+   k_5 &= f\!\Bigl(\mathbf{x}_n + \Delta t\bigl(\tfrac{1}{2}k_1 - \tfrac{3}{2}k_3 + 2k_4\bigr),\;t_n + \tfrac{2\Delta t}{3}\Bigr),\\
+   k_6 &= f\!\Bigl(\mathbf{x}_n + \Delta t\bigl(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4\bigr),\;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).
+
+Implementation steps:
+
+  1. Compute each :math:`k_i = \sigma\,\mathbf{u}(\cdot)` at its intermediate point.  
+  2. Form the weighted sum:  
+     .. math::  
+        \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)
+
+All methods assume a continuous, differentiable velocity field via tricubic interpolation; replacing :math:`f` by the chosen sampler affects only boundary‐condition treatment.
+
+
+.. _ftlefinal:
+
+FTLE Computation
+-------------------
 
 
 

source/5_gui.rst

@@ -3,4 +3,8 @@
 Graphical User Interface (GUI)
 ===============
 
-This page describes
+This page describes how to use GUI mode of ``PyFTLE3D``, which is friendly for who would like to use the software without command-line operations.
+The author holds the opinion that coding should not be a barrier for using scientific analysis tools such as ``FTLE``.
+For some peers or even experts and professors, coding is absolutely boring.
+
+The GUI shares exactly the same functionalities as the command-line mode, but is more difficult to customize, as ``PyQt`` is used for building it.