Lax–Wendroff method

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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.

[edit] References

P.D Lax; B. Wendroff (1960). "Systems of conservation laws". Commun. Pure Appl Math. 13: 217–237.
Michael J. Thompson, An Introduction to Astrophysical Fluid Dynamics, Imperial College Press, London, 2006.