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 $x = \pm \infty$. We show that the solutions tend to the discrete diffusion waves at the rate $O({t^{ - 3/4 + 1/2p + \sigma }})$ in ${l^p}$, $1 \leq p \leq \infty$, with $\sigma > 0$ 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.