Asymptotic stability of discrete shock waves for the Lax-Friedrichs scheme to hyperbolic systems of conservation laws

Abstract

Let φ ⁿ_j be a nonstationary discrete shock wave of the Lax-Friedrichs scheme to a hyperbolic system of conservation laws, andu ^o_j be a perturbation of φ ^o_j . We study the asymptotic behavior of the discrete solution u ⁿ_j of the Lax-Friedrichs scheme with initial datau ^o_j . Without the restriction of zero total perturbation, we prove that if the strength of the shock is small and the perturbation is small too, then on the even mesh u ⁿ_j − φ ⁿ_j (ξ′) tends to zero in thel ² norm asn → ∞, where φ ⁿ_j (ξ′) is a shifted discrete shock wave. This result on the odd mesh is the same with respect to another discrete shock wave.