Large-time behavior of solutions of Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws

Abstract

We study the large-time behavior of discrete solutions of the Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws. The initial data considered here are small and tend to a constant state at x = ± ∞ x = \pm \infty . We show that the solutions tend to the discrete diffusion waves at the rate O ( t − 3 / 4 + 1 / 2 p + σ ) O({t^{ - 3/4 + 1/2p + \sigma }}) in l p {l^p} , 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , with σ > 0 \sigma > 0 being an arbitrarily small constant. The discrete diffusion waves can be constructed from the self-similar solutions of the heat equation and the Burgers equation through an averaging process.