Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes

The nonlinear stability of travelling Lax shocks in semidiscrete conservation laws involving general spatial forward-backward discretization schemes is considered. It is shown that spectrally stable semidiscrete Lax shocks are nonlinearly stable. In addition, it is proved that weak semidiscrete Lax profiles satisfy the spectral stability hypotheses made here and are therefore nonlinearly stable. The nonlinear stability results are proved by constructing the resolvent kernel using exponential dichotomies, which have recently been developed in this setting, and then using the contour integral representation for the associated Green’s function to derive pointwise bounds that are sufficient for proving nonlinear stability. Previous stability analyses for semidiscrete shocks relied primarily on Evans functions, which exist only for one-sided upwind schemes. ∗Beck was partially supported under NSF grant DMS-0602891 †Hupkes acknowledges support from the Netherlands Organization for Scientific Research (NWO) ‡Sandstede was partially supported by the NSF through grant DMS-0907904 §Zumbrun was partially supported by the NSF through grant DMS-0801745