Convergence rates to discrete shocks for nonconvex conservation laws

Abstract

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 \(j\rightarrow \pm \infty\), respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over \((-\infty, j)\) is small and decays with an algebraic rate as \(|j|\rightarrow \infty\), then the perturbations to discrete shocks are shown to decay with the corresponding rate as \(n\rightarrow \infty\). 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.