Decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws

This paper is to study the decay rate for perturbations of stationary discrete shocks for the Lax-Friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method. If the summation of the initial perturbation over $(-\infty , j)$ is small and decays at the algebraic rate as $|j|\rightarrow \infty$, then the solution approaches the stationary discrete shock profiles at the corresponding rate as $n\rightarrow \infty$. This rate seems to be almost optimal compared with the rate in the continuous counterpart. Proofs are given by applying the weighted energy integration method to the integrated scheme of the original one. The selection of the time-dependent discrete weight function plays a crucial role in this procedure.