Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes

Margaret Beck, Hermen Jan Hupkes, Bjoern Sandstede, Kevin Zumbrun

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

The nonlinear stability of traveling 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.

Original languageEnglish
Pages (from-to)857-903
Number of pages47
JournalSIAM Journal on Mathematical Analysis
Volume42
Issue number2
DOIs
Publication statusPublished - 2010

Keywords

  • lattice dynamical system
  • semidiscrete conservation law
  • shocks
  • nonlinear stability
  • exponential dichotomies
  • FUNCTIONAL-DIFFERENTIAL EQUATIONS
  • CONSERVATION-LAWS
  • DISCRETE SHOCKS
  • MIXED-TYPE
  • EXPONENTIAL DICHOTOMIES
  • FUNCTION BOUNDS
  • PROFILES
  • SYSTEMS
  • WAVES
  • APPROXIMATIONS

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