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Next: 5.1.3 Leapfrog Scheme (LF) Up: 5.1 Initial Value Problems Previous: 5.1.1 FTCS Scheme; Stability


5.1.2 Lax Scheme

Replacing in the FTCS formula the term $\mbox{$\bf u$}_{j}^{n}$ by its spatial average $[\mbox{$\bf u$}_{j+1}^{n}+\mbox{$\bf u$}_{j-1}^{n}]/2$, we approximate $\partial \mbox{$\bf u$}/\partial t = - \partial \mbox{$\bf j$}/\partial x$ by
    $\displaystyle \fbox{$ \displaystyle
\mbox{$\bf u$}_{j}^{n+1} = \frac{1}{2}
\lef...
...2 \Delta x}
\left[ \mbox{$\bf j$}_{j+1}^{n}-\mbox{$\bf j$}_{j-1}^{n} \right]
$}$  



\begin{figure}\includegraphics[width=120pt]{figures/f5lax.ps}
\end{figure}
Stability / Friedrichs-Löwy condition:

Insert Fourier expanded $u(x)$ in Lax formula to find
$\displaystyle g(k)= \cos \, k \Delta x -i \frac{c \Delta t}{\Delta x} \sin \, k \Delta x$      

The condition $\vert g(k)\vert \leq 1$ is tantamount to
    $\displaystyle \fbox{ $\displaystyle
\frac{\vert c\vert \Delta t}{\Delta x} \leq 1 $
}$  

\begin{figure}\includegraphics[width=120pt]{figures/f5cfl.ps}
\end{figure}
Region below the dashed line: physically relevant for $u_{j}^{n+1}$, according to $x(t_{n+1})=x(t_{n})\pm \vert c\vert \, \Delta t$

Close scrutiny shows that LAX solves not the original PDE but
$\displaystyle \frac{\partial u}{\partial t}$ $\textstyle =$ $\displaystyle -c \frac{\partial u}{\partial x}
+ \frac{(\Delta x)^{2}}{2 \Delta t} \frac{\partial^{2} u}{\partial x^{2}}$  

The additional diffusive term makes the method stable. However, it is an artefact and should be small:
$\displaystyle \vert c\vert \Delta t$ $\textstyle >>$ $\displaystyle \frac{\Delta x}{2}\,\frac{\vert\delta^{2} u\vert}{\vert\delta u\vert}$  


next up previous
Next: 5.1.3 Leapfrog Scheme (LF) Up: 5.1 Initial Value Problems Previous: 5.1.1 FTCS Scheme; Stability
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001