Lax–Wendroff method

The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite differences. It is second-order accurate in both space and time.

Suppose one has an equation of the following form:

: $frac\{partial\; f(x,t)\}\{partial\; t\}=frac\{partial\; g(f(x,t))\}\{partial\; x\},$

where "x" and "t" are independent variables, and the initial state, ƒ("x", 0) is given.

The first step in the Lax–Wendroff method calculates values for ƒ("x", "t") at half time steps, "t"_{"n" + 1/2} and half grid points, "x"_{"i" + 1/2}. In the second step values at "t"_{"n" + 1} are calculated using the data for "t"_"n" and "t"_{"n" + 1/2}.

First (Lax) step:

: $cfrac\{f\_\{i+1/2\}^\{n+1/2\}\; -\; cfrac\{f\_i^n+f\_\{i+1\}^n\}\{2\{(1/2)\; *\; Delta\; t\}=cfrac\{g\_\{i+1\}^n\; -\; g\_i^n\}\{Delta\; x\}.,$

Second step:

: $cfrac\{f\_i^\{n+1\}\; -\; f\_i^n\}\{Delta\; t\}=cfrac\{g\_\{i+1/2\}^\{n+1/2\}\; -\; g\_\{i-1/2\}^\{n+1/2\{Delta\; x\}.,$

This method can be further applied to some systems of partial differential equations.

References

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* Michael J. Thompson, "An Introduction to Astrophysical Fluid Dynamics", Imperial College Press, London, 2006.