Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws

TL;DR

This paper proves the nonlinear orbital stability of spectrally stable discrete shock profiles for scalar conservation laws in weighted spaces, without amplitude restrictions, and applicable to schemes with artificial viscosity.

Contribution

It establishes nonlinear stability results for a broad class of schemes without amplitude restrictions, extending previous work and providing a foundation for systems of conservation laws.

Findings

Spectral stability implies nonlinear stability in weighted spaces.

Applicable to schemes with artificial high-order viscosity.

Provides a framework for extending to systems of conservation laws.

Abstract

For scalar conservation laws, we prove that spectrally stable stationary Lax discrete shock profiles are nonlinearly stable in some polynomially-weighted $\ell^1$ and $\ell^\infty$ 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.