Lax–Wendroff method

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:

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

In the linear case, where f(u) = Au , and A is a constant,

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

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

where ${\displaystyle A_{i\pm 1/2}}$ is the Jacobian matrix evaluated at ${\displaystyle {\frac {1}{2}}(U_{i}^{n}+U_{i\pm 1}^{n})}$.

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