Existence and asymptotic stability of relaxation discrete shock profiles

Abstract

Abstract. In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in L 2 , provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

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

“…Finally, let me point to different but somewhat related problems: selfsimilar profiles for the Broadwell model with a relaxation time of the form 1/ε t in [56], existence of discrete traveling waves for relaxing schemes in [57][58][59], non-autonomous case, motivated by traffic modeling in the presence of a moving barrier which blocks a single lane, in [60].…”

“…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).…”