Existence and stability of traveling discrete shocks

Abstract

We are concerned with existence and stability questions for finite difference schemes approximating solutions of scalar conservation laws with shocks. A suitable model for the study of the artifacts created by these schemes near the shocks are the traveling discrete shock profiles; these are discrete shock profiles υ=(υk)k∈ℤ which reappear shifted, when the scheme is applied on them, according to the speed of the shock. Existence of such profiles connecting entropy admissible shocks is already established for monotone schemes, first and third order accurate schemes and the Lax-Wendroff scheme. Jennings showed existence and l1-stability of these profiles for conservative monotone schemes. Smyrlis showed existence and parametrization by the amount of excess mass and stability for stationary profiles of the Lax-Wendroff scheme. Shih Hsien Yu showed existence of traveling profiles of mild strength for the Lax-Wendroff scheme using inertial manifolds theory. Here we study traveling discrete shock profiles for Lax-Wendroff, Engquist-Osher and monotone schemes. We show existence of such profiles with small shock speed. We also show that these profiles are stable with respect to suitably weighted l2-norms.