Update 3_numerical.rst

Author: hweifluids

source/3_numerical.rst

@@ -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