Lax–Wendroff method explained

The Lax–Wendroff method, named after Peter Lax and Burton Wendroff,^[1] 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.

Definition

Suppose one has an equation of the following form:$$\backslash frac\; +\; \backslash frac\; =\; 0$$where and are independent variables, and the initial state, is given.

Linear case

In the linear case, where, and is a constant,^[2] $$u\_i^\; =\; u\_i^n\; -\; \backslash frac\; A\backslash left[u\_\{i+1\}^\{n\}\; -\; u\_\{i-1\}^\{n\}\; \backslash right]\; +\; \backslash frac\; A^2\backslash left[u\_\{i+1\}^\{n\}\; -2\; u\_\{i\}^\{n\}\; +\; u\_\{i-1\}^\{n\}\; \backslash right].$$Here

n refers to the t dimension and i refers to the x dimension.This linear scheme can be extended to the general non-linear case in different ways. One of them is letting$$A(u)\; =\; f\text{'}(u)\; =\; \backslash frac$$

Non-linear case

The conservative form of Lax-Wendroff for a general non-linear equation is then:$$u\_i^\; =\; u\_i^n\; -\; \backslash frac\; \backslash left[f(u\_\{i+1\}^\{n\})\; -\; f(u\_\{i-1\}^\{n\})\; \backslash right]\; +\; \backslash frac\; \backslash left[A\_\{i+1/2\}\; \backslash left(f(u\_\{i+1\}^\{n\})\; -\; f(u\_\{i\}^\{n\})\backslash right)\; -\; A\_\{i-1/2\}\backslash left(f(u\_\{i\}^\{n\})-f(u\_\{i-1\}^\{n\})\backslash right)\; \backslash right].$$where

A _{i\pm} is the Jacobian matrix evaluated at $\backslash frac\; (u^n\_i\; +\; u^n\_)$.

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 at half time steps, and half grid points, . In the second step values at are calculated using the data for and .

First (Lax) steps:$$u\_^\; =\; \backslash frac(u\_^n\; +\; u\_^n)\; -\; \backslash frac(f(u\_^n)\; -\; f(u\_^n)),$$$$u\_^=\; \backslash frac(u\_^n\; +\; u\_^n)\; -\; \backslash frac(f(u\_^n)\; -\; f(u\_^n)).$$

Second step:$$u\_i^\; =\; u\_i^n\; -\; \backslash frac\; \backslash left[f(u\_\{i+1/2\}^\{n+1/2\})\; -\; f(u\_\{i-1/2\}^\{n+1/2\})\; \backslash right].$$

MacCormack method

See main article: 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:$$u\_^=\; u\_^n\; -\; \backslash frac(f(u\_^n)\; -\; f(u\_^n)).$$Second step:$$u\_i^\; =\; \backslash frac\; (u\_^n\; +\; u\_^*)\; -\; \backslash frac\; \backslash left[f(u\_\{i\}^\{*\})\; -\; f(u\_\{i-1\}^\{*\})\; \backslash right].$$

Alternatively,First step:$$u\_^\; =\; u\_^n\; -\; \backslash frac(f(u\_^n)\; -\; f(u\_^n)).$$Second step:$$u\_i^\; =\; \backslash frac\; (u\_^n\; +\; u\_^*)\; -\; \backslash frac\; \backslash left[f(u\_\{i+1\}^\{*\})\; -\; f(u\_\{i\}^\{*\})\; \backslash right].$$

References

• Michael J. Thompson, An Introduction to Astrophysical Fluid Dynamics, Imperial College Press, London, 2006.
• Book: Press . WH . Teukolsky . SA . Vetterling . WT . Flannery . BP . 2007 . Numerical Recipes: The Art of Scientific Computing . 3rd . Cambridge University Press . New York . 978-0-521-88068-8 . Section 20.1. Flux Conservative Initial Value Problems . http://apps.nrbook.com/empanel/index.html#pg=1040 . 1040.

Notes and References

1. P.D Lax . B. Wendroff . 1960 . Systems of conservation laws . Commun. Pure Appl. Math. . 13 . 217–237 . 10.1002/cpa.3160130205 . 2 . https://web.archive.org/web/20170925220837/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA385056 . live . September 25, 2017 .
2. Book: LeVeque, Randall J. . Numerical Methods for Conservation Laws . Boston . Birkhäuser . 1992 . 0-8176-2723-5 . 125 .