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arXiv · 2409.18930

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

Lucas Coeuret
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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.

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Lucas Coeuret. 2024-09-27. Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws. https://arxiv.org/abs/2409.18930

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