time integration

Author: hweifluids

source/3_numerical.rst

@@ -25,102 +25,70 @@ Wall Treatment
 Time Integration
 ~~~~~~~~~~~~~~~~~~
 
-Consider the initial-value problem for particle advection in a velocity field
+Consider the initial‐value problem for passive tracer advection in a continuous velocity field
 
 .. math::
 
-   \frac{d\mathbf{x}}{dt} = \sigma\,\mathbf{u}(\mathbf{x},t),
-   \quad \mathbf{x}(t_n) = \mathbf{x}_n,
+   \frac{d\mathbf{x}}{dt} = \sigma\,\mathbf{u}(\mathbf{x},t)\,,  
+   \mathbf{x}(t_n)=\mathbf{x}_n\,,  
 
-where :math:`\sigma = \pm 1` indicates forward/backward advection.
+where :math:`\sigma = \pm1` selects forward or backward integration.
 
 **Explicit Euler Method**
 
-Given :math:`\mathbf{x}_n` at time :math:`t_n`, the first-order (Euler) update is
+The first‐order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:
 
 .. 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)
+   \mathbf{u}_n = \mathbf{u}(\mathbf{x}_n,t_n),\\
+   \mathbf{x}_{n+1} = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}_n.
 
-Implementation steps:
+This method incurs a global error of order :math:`O(\Delta t)` and requires only one velocity evaluation per step.
 
-  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  
+**Second‐Order Runge–Kutta (Heun’s Method)**
 
-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:
+Heun’s method attains second‐order accuracy by combining predictor and corrector slopes:
 
 .. 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:
+   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).
 
-  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)
+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)**
 
-A four-stage scheme with :math:`O(\Delta t^4)` accuracy:
+The classical RK4 method achieves fourth‐order accuracy via four slope evaluations at intermediate points:
 
 .. 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).
+   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).
 
-Implementation steps:
+This yields a global error of order :math:`O(\Delta t^4)` with four velocity evaluations per step.
 
-  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)**
 
-**Sixth-Order Runge–Kutta (RK6)**
-
-A six-stage explicit method with :math:`O(\Delta t^6)` accuracy. Define:
+The six‐stage scheme uses non‐uniform weights to attain sixth‐order accuracy:
 
 .. 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)
+   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).
 
-All methods assume a continuous, differentiable velocity field via tricubic interpolation; replacing :math:`f` by the chosen sampler affects only boundary‐condition treatment.
+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:
 

source/5_gui.rst

@@ -5,6 +5,6 @@ Graphical User Interface (GUI)
 
 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.
+For some peers or even experts and professors, coding is absolutely boring, especially for GPU tasks on such large dataset scale.
 
 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.