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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-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 up previous
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