Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws
<|reference_start|>Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws: 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 [Coeu23]. 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.<|reference_end|>
@article{coeuret2024nonlinear,
    title={Nonlinear orbital stability of stationary discrete shock profiles for
  scalar conservation laws},
    author={Lucas Coeuret},
    journal={arXiv preprint arXiv:2409.18930},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.18930},
    primaryClass={math.AP cs.NA math.NA}
}