Convergence rates to discrete shocks for nonconvex conservation laws

This paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws, We assume that the discrete initial data tend to constant states as j --> +/- infinity, respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over (- infinity , j) is small and decays with an algebraic rate as /j/ --> infinity, then the perturbations to discrete shocks are shown to decay with the corresponding rate as n --> infinity. The proof is given by applying weighted energy estimates. A discrete weight function, which depends on the space-time variables for the decay rate and the state of the discrete shocks in order to treat the non-convexity, plays a crucial role.