2024/09/27 by Lucas Coeuret, Coeuret, Lucas
Mathematics · Physics and Astronomy · #35L65 #65M06 #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2409.18930
openalex publication_date 2024/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For scalar conservation laws, we prove that spectrally stable stationary Lax discrete shock profiles are nonlinearly stable in some polynomially-weighted ℓ1 and ℓ^∞ spaces. In comparison with several previous nonlinear stability results on discrete shock profiles, we avoid the introduction of any weakness assumption on the amplitude of the shock and apply our analysis to a large family of schemes that introduce some artificial possibly high-order viscosity. The proof relies on a precise description of the Green's function of the linearization of the numerical scheme about spectrally stable discrete shock profiles obtained in [Coeu25]. The present article also pinpoints the ideas for a possible extension of this nonlinear orbital stability result for discrete shock profiles in the case of systems of conservation laws.