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

Definition[edit]

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[edit]

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[edit]

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

where is the Jacobian matrix evaluated at .

Jacobian free methods[edit]

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

Richtmyer method[edit]

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(xt)) at half time steps, tn + 1/2 and half grid points, xi + 1/2. In the second step values at tn + 1 are calculated using the data for tn and tn + 1/2.

First (Lax) steps:

Second step:

MacCormack method[edit]

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[edit]

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