Two-step Lax-Friedrichs method

The Lax-Friedrichs (LxF) method [2, 3, 4] is a basic method for the solution of hyperbolic partial differential equations (PDEs). Its use is limited because its order is only one, but it is easy to program, applicable to general PDEs, and has good qualitative properties because it is monotone. The LxF method is often used to show the effects of dissipation, but it is not actually a dissipative method, a point made by Strikwerda [3, pp. 100–101]. The amplification factor |g(θ)| is 1 for θ = π, so the highest frequency oscillation on the mesh is not damped at all and other high frequencies are weakly damped. Example 7.8.3 of [4] provides a concrete example. At the conclusion of his illuminating discussion of the example, Thomas writes “And, finally, the appearance of the highly oscillatory mode is a part of the Lax-Friedrichs scheme. There is no easy way to get rid of it.” In this note we describe an easy way to do exactly that. The solution u(x, t) of a PDE is approximated on a mesh with constant increments ∆x in space and ∆t in time: v m ≈ u(m∆x, n∆t). For the one-way wave equation ut + aux = 0, the LxF method is