Convergence rates to discrete shocks for nonconvex conservation laws

Summary.

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.