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.

Suppose one has an equation of the following form:

 \frac{\partial u(x,t)}{\partial t}=\frac{\partial f(u(x,t))}{\partial x}\,

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

The first step in the Lax–Wendroff method calculates values for 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:

 u_{i+1/2}^{n+1/2} = \frac{1}{2}(u_{i+1}^n + u_{i}^n) - \frac{\Delta t}{2\,\Delta x}( f( u_{i+1}^n ) - f( u_{i}^n ) ),
 u_{i-1/2}^{n+1/2} = \frac{1}{2}(u_{i}^n + u_{i-1}^n) - \frac{\Delta t}{2\,\Delta x}( f( u_{i}^n ) - f( u_{i-1}^n ) ).

Second step:

 u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x} \left[ f(u_{i+1/2}^{n+1/2}) - f(u_{i-1/2}^{n+1/2}) \right].

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