The Lax–Wendroff method, named after

Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dy ...

and Burton Wendroff, is a numerical method for the solution of

hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can ...

s, based on

finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

s. 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:

\frac + \frac = 0

where and are independent variables, and the initial state, is given.

Linear case

In the linear case, where , and is a constant,

u_i^ = u_i^n - \frac A\left u_^ - u_^ \right + \frac A^2\left u_^ -2 u_^ + u_^ \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'(u) = \frac

Non-linear case

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

u_i^ = u_i^n - \frac \left f(u_^) - f(u_^) \right + \frac \left A_ \left(f(u_^) - f(u_^)\right) - A_\left( f(u_^)-f(u_^)\right) \right

where

A_

is the Jacobian matrix evaluated at

\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_^ = \frac(u_^n + u_^n) - \frac( f(u_^n) - f(u_^n) ),

u_^= \frac(u_^n + u_^n) - \frac( f(u_^n) - f(u_^n) ).

Second step:

u_i^ = u_i^n - \frac \left f(u_^) - f(u_^) \right

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 - \frac( f(u_^n) - f(u_^n) ).

Second step:

u_i^ = \frac (u_^n + u_^*) - \frac \left f(u_^) - f(u_^) \right

Alternatively, First step:

u_^ = u_^n - \frac( f(u_^n) - f(u_^n)  ).

Second step:

u_i^ = \frac (u_^n + u_^*) - \frac \left f(u_^) - f(u_^) \right

References

* Michael J. Thompson, ''An Introduction to Astrophysical Fluid Dynamics'', Imperial College Press, London, 2006. * {{DEFAULTSORT:Lax-Wendroff Method Numerical differential equations Computational fluid dynamics