A systematic approach for correcting nonlinear instabilities - The Lax-Wendroff scheme for scalar conservation laws

The Lax-Wendroff (L-W) difference scheme for a single conservation law has been shown to be nonlinearly unstable near stagnation points. In this paper a simple variant of L-W is devised which has all the usual properties-conservation form, three point scheme, second order accurate on smooth solutions, but which is shown rigorously to be L² stable for Burger's equation and which is believed to be stable in general. This variant is constructed by adding a simple nonlinear viscosity term to the usual L-W operator. The nature of the viscosity is deduced by first stabilizing, for general conservation laws, a model differential equation derived by analyzing the truncation error for L-W, keeping only terms of order (Δt)². The same procedure is then carried out for an analogous semi-discrete model. Finally, the full L-W difference scheme is rigorously shown to be stable provided that the C.F.L. restriction λ|u_jⁿ|≦0.24 is satisfied.