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:

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

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:

${\displaystyle {\cfrac {f_{i+1/2}^{n+1/2}-{\cfrac {f_{i}^{n}+f_{i+1}^{n}}{2}}}{(1/2)*\Delta t}}={\cfrac {g_{i+1}^{n}-g_{i}^{n}}{\Delta x}}.\,}$

Second step:

${\displaystyle {\cfrac {f_{i}^{n+1}-f_{i}^{n}}{\Delta t}}={\cfrac {g_{i+1/2}^{n+1/2}-g_{i-1/2}^{n+1/2}}{\Delta x}}.\,}$

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

References

• P.D Lax (1960). "Systems of conservation laws". Commun. Pure Appl Math. 13: 217–237. doi:10.1002/cpa.3160130205. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Michael J. Thompson, An Introduction to Astrophysical Fluid Dynamics, Imperial College Press, London, 2006.