5.1.5 Lax and Lax-Wendroff in Two Dimensions

Next: 5.1.6 Resumé: Conservative-hyperbolic DE Up: 5.1 Initial Value Problems Previous: 5.1.4 Lax-Wendroff Scheme (LW)

5.1.5 Lax and Lax-Wendroff in Two Dimensions

$\begin{displaymath} \frac{\partial u}{\partial t} = -\frac{\partial j_{x}}{\partial x} -\frac{\partial j_{y}}{\partial y} \end{displaymath}$

(advective case: $j_{x}=c_{x}u$ and $\,j_{y}=c_{y}u$ )

Lax scheme:

$\begin{displaymath} u_{i,j}^{n+1} = \frac{1}{4} \left[ u_{i+1,j}^{n}+u_{i,j+1}^... ...a t}{2 \Delta y} \left[j_{y,i,j+1}^{n}-j_{y,i,j-1}^{n}\right] \end{displaymath}$

**Figure 5.2:** Lax method in two dimensions
$\begin{figure}\includegraphics[width=120pt]{figures/f5lx2d.ps} \end{figure}$

Lax-Wendroff:

For the second stage (half-step leapfrog) we need $j_{x,i+1/2,j-1/2}^{n+1/2}$ etc., which requires $u_{i+1/2,j-1/2}^{n+1/2}$ , which must be determined from $u_{i,j-1/2}^{n}$ , $u_{i+1,j-1/2}^{n}$ etc.
But: quantities with half-step spatial indices ( $\scriptstyle i+1/2$ , $\scriptstyle j-1/2$ etc.) are given at half-step times ( $t_{n+1/2}$ ) only.
Modifying the LW scheme to allow for this, we have Lax-Wendroff in 2 dimensions:

Lax method to determine the -values at half-step time $t_{n+1/2}$ :

$\displaystyle u_{i+1,j}^{n+1/2}$ $\textstyle =$ $\displaystyle \frac{1}{4} \left[ u_{i+2,j}^{n}+u_{i+1,j+1}^{n}+u_{i,j}^{n}+u_{i+1,j-1}^{n}\right]$

$\displaystyle \;\;\;\; -\frac{\Delta t}{2\Delta x} \left[ j_{x,i+2,j}^{n}-j_{x,i,j}^{n}\right]$

$\displaystyle \;\;\;\; -\frac{\Delta t}{2\Delta y} \left[ j_{y,i+1,j+1}^{n}-j_{y,i+1,j-1}^{n}\right]$

etc.
Evaluation at half-step time:

$\begin{displaymath} u_{i+1,j}^{n+1/2}\, , \dots \; \Longrightarrow \; j_{x,i+1,j}^{n+1/2}\, , \dots \end{displaymath}$
leapfrog with half-step:

$\displaystyle u_{i,j}^{n+1}=u_{i,j}^{n}$ $\textstyle -$ $\displaystyle \frac{\Delta t}{2\Delta x} \left[ j_{x,i+1,j}^{n+1/2}-j_{x,i-1,j}^{n+1/2} \right]$

$\textstyle -$ $\displaystyle \frac{\Delta t}{2\Delta y} \left[ j_{y,i,j+1}^{n+1/2}-j_{y,i,j-1}^{n+1/2} \right]$

Lax-Wendroff in two dimensions

**Figure:** First stage (= Lax) in the 2-dimensional LW method: $\circ$ ... $t_{n},t_{n+1}$ , $\Box$ ... $t_{n+1/2}$
$\begin{figure}\includegraphics[width=120pt]{figures/f5lw2d2.ps}\end{figure}$

For $u_{i,j}^{n+1}$ only the points $\circ$ (at $t_{n}$ ) are used;
for $u_{i+1,j}^{n+1}$ we use the points $\Box$ .

Problem: Drift between subgrids $\circ$ and $\Box$ .

Solution: If the given PDE contains a diffusive term, this guarantees coupling. Otherwise, artificially add a small diffusive term.

Stability analysis: Fourier modes are now 2-dimensional:

$\begin{displaymath} u(x,y)=\sum_{k} \sum_{l} U_{k,l} e^{ikx+ily} \end{displaymath}$

Assuming $\Delta x = \Delta y$ we find the CFL condition

$\begin{displaymath} \Delta t \leq \frac{\Delta x}{\sqrt{2} \, \sqrt{c_{x}^{2}+c_{y}^{2}}} \end{displaymath}$

Next: 5.1.6 Resumé: Conservative-hyperbolic DE Up: 5.1 Initial Value Problems Previous: 5.1.4 Lax-Wendroff Scheme (LW)

Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001

$\displaystyle u_{i+1,j}^{n+1/2}$	$\textstyle =$	$\displaystyle \frac{1}{4} \left[ u_{i+2,j}^{n}+u_{i+1,j+1}^{n}+u_{i,j}^{n}+u_{i+1,j-1}^{n}\right]$
		$\displaystyle \;\;\;\; -\frac{\Delta t}{2\Delta x} \left[ j_{x,i+2,j}^{n}-j_{x,i,j}^{n}\right]$
		$\displaystyle \;\;\;\; -\frac{\Delta t}{2\Delta y} \left[ j_{y,i+1,j+1}^{n}-j_{y,i+1,j-1}^{n}\right]$

$\displaystyle u_{i,j}^{n+1}=u_{i,j}^{n}$	$\textstyle -$	$\displaystyle \frac{\Delta t}{2\Delta x} \left[ j_{x,i+1,j}^{n+1/2}-j_{x,i-1,j}^{n+1/2} \right]$
	$\textstyle -$	$\displaystyle \frac{\Delta t}{2\Delta y} \left[ j_{y,i,j+1}^{n+1/2}-j_{y,i,j-1}^{n+1/2} \right]$