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5.1.5 Lax and LaxWendroff in Two Dimensions
(advective case: and
)
Lax scheme:
Figure 5.2:
Lax method in two dimensions

LaxWendroff:
For the second stage (halfstep leapfrog) we need
etc., which requires
, which must be determined from
,
etc.
But: quantities with halfstep spatial indices
(
,
etc.)
are given at halfstep times
() only.
Modifying the LW scheme to allow for this, we have
LaxWendroff in 2 dimensions:
 Lax method to determine the values at halfstep time
:
etc.
 Evaluation at halfstep time:
 leapfrog with halfstep:
LaxWendroff in two dimensions
Figure:
First stage (= Lax) in the 2dimensional LW method:
...
, ...

For only the points
(at ) are
used;
for
we use the points .
Problem: Drift between subgrids and .
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 2dimensional:
Assuming
we find the CFL condition
Next: 5.1.6 Resumé: Conservativehyperbolic DE
Up: 5.1 Initial Value Problems
Previous: 5.1.4 LaxWendroff Scheme (LW)
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001