Existence and stability of stationary profiles of the lw scheme

In this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws When a difference scheme introduces artificial numerical diffusion, for example the Lax-Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax-Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock. In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first-order accurate; they are known to be contractive in the sense of any l^p norm. Jennings showed existence and l¹ stability of traveling discrete smeared shocks for such schemes. Here we study similar questions for the Lax-Wendroff scheme without artificial viscosity; this is a nonmonotone, second-order accurate scheme. We prove existence of a one-parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l² norm. The proof relies on studying the linearized Lax-Wendroff scheme.