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Next: 5.2 Initial Value Problems Up: 5.1 Initial Value Problems Previous: 5.1.5 Lax and Lax-Wendroff

5.1.6 Resumé: Conservative-hyperbolic DE

$\displaystyle \frac{\partial \mbox{$\bf u$}}{\partial t}$ $\textstyle =$ $\displaystyle - \frac{\partial \mbox{$\bf j$}}{\partial x}$  

- Use Lax-Wendroff!

- If not, use at least Lax, but see that in addition to CFL
$\displaystyle \vert c\vert \Delta t$ $\textstyle >>$ $\displaystyle \frac{\Delta x}{2}\,\frac{\vert\delta^{2} u\vert}{\vert\delta u\vert}$  

- Forget FTCS and Leapfrog!

To test the various methods, let us apply them to the 1D wave equation. When altering the propagation velocity $c$, the time step $\Delta t$, or the grid width $\Delta x$, keep in mind the operation regions of the different algorithms:

FTCS - unstable
LAX, LEAPFROG, LAX-WENDROFF - CFL condition $ \vert c\vert \Delta t / \Delta x \leq 1 $

Applet Hpde1: Start

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