Convergence rates to the discrete travelling wave for relaxation schemes

Abstract

This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time, provided the sums of the initial perturbations for the u-component equal zero. A polynomially weighted l 2 norm on the perturbation of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.

Twenty-eight years with “Hyperbolic conservation laws with relaxation”

“…To complete the outlook on the stability problem, it is worthy to quote [58,59,85] on the stability of discrete relaxation shocks, [54,55] relative to the case of compresence of diffusion and relaxation, [86] dealing with the stability of contact discontinuities for the Jin-Xin model, and [87] concerning a specific model where stability of large discontinuous relaxation shocks can be proved. The last paper is one of the few concerning stability in presence of a jump, but it should be stressed that model is very specific since it can be reduced to a triangular system and thus, essentially, to a problem in scalar conservation laws (with source).…”

Existence and asymptotic stability of relaxation discrete shock profiles

“…There are several authors who have studied the stability of shock profiles; see [2,3,5,6,25,26]. In the study of discrete relaxation conservation laws, most works have concentrated on the scalar case, such as Liu, Wang and Yang [13,14,15]. In this paper, we will concentrate on the case of systems.…”