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Next: 5.1.5 Lax and Lax-Wendroff Up: 5.1 Initial Value Problems Previous: 5.1.3 Leapfrog Scheme (LF)


5.1.4 Lax-Wendroff Scheme (LW)

\begin{figure}\includegraphics[width=120pt]{figures/f5lw.ps}
\end{figure}


\fbox{
\begin{minipage}{450pt}
\begin{itemize}
\item Lax method with half-step: ...
...}_{j-1/2}^{n+1/2} \right]
\nonumber \end{eqnarray}\end{itemize}\end{minipage}
}

Stability:

Once more assuming $j=cu$ and using the ansatz $U_{k}^{n+1}= g(k) U_{k}^{n}$ we find
$\displaystyle g(k)$ $\textstyle =$ $\displaystyle 1-ia \, \sin \, k \Delta x -a^{2}(1-\cos \, k \Delta x),$  

with $a=c \Delta t/ \Delta x$. The requirement $\vert g\vert^{2}\leq 1$ leads once again to the CFL condition, $
c\,\Delta t/\Delta x \leq 1
$.



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