Lax-Friedrichs method

The Lax-Friedrichs method is a numerical scheme for solving hyperbolic partial differential equations by finite differences, named after the mathematicians Peter Lax and Kurt O. Friedrichs. It can be read as the forward-in-time, centred-in-space scheme with a numerical dissipation term of one half added, and it offers an alternative to Godunov's scheme because it avoids solving a Riemann problem at each cell interface, paying for that simplicity with artificial viscosity. The method is explicit and first order accurate in both time and space, and its generalisation to nonlinear conservation laws is conservative but quite dissipative. A von Neumann stability analysis shows that it is stable only when a condition relating the time step to the grid spacing is satisfied. Because it smears discontinuities strongly, it is mostly valued as a building block for higher order schemes, much as Euler time steps underpin higher order integrators for ordinary differential equations.