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. This method is an example of explicit time integration where the function that defines the governing equation is evaluated at the current time.

## Contents

## Definition

Suppose one has an equation of the following form:

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

### Linear case

In the linear case, where * f(u) = Au *, and *A* is a constant,^{[1]}

This linear scheme can be extended to the general non-linear case in different ways. One of them is letting

### Non-linear case

The conservative form of Lax-Wendroff for a general non-linear equation is then:

where is the Jacobian matrix evaluated at .

## Jacobian free methods

To avoid the Jacobian evaluation, use a two-step procedure.

### Richtmyer method

What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(*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) steps:

Second step:

### MacCormack method

Another method of this same type was proposed by MacCormack. MacCormack's method uses first forward differencing and then backward differencing:

First step:

Second step:

Alternatively, First step:

Second step:

## References

**^**LeVeque, Randy J.*Numerical Methods for Conservation Laws", Birkhauser Verlag, 1992, p. 125.*

- P.D Lax; B. Wendroff (1960). "Systems of conservation laws".
*Commun. Pure Appl. Math*.**13**(2): 217–237. doi:10.1002/cpa.3160130205. - Michael J. Thompson,
*An Introduction to Astrophysical Fluid Dynamics*, Imperial College Press, London, 2006. - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 20.1. Flux Conservative Initial Value Problems".
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.). New York: Cambridge University Press. p. 1040. ISBN 978-0-521-88068-8.