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.

翻译：对于标量守恒律，我们证明了谱稳定的平稳Lax离散激波剖面在某些多项式加权的$\ell^1$和$\ell^\infty$空间中是非线性稳定的。与先前若干关于离散激波剖面的非线性稳定性结果相比，我们避免了对激波振幅引入任何弱性假设，并将分析应用于一大类引入人工（可能高阶）粘性的数值格式。证明依赖于[Coeu23]中获得的、关于谱稳定离散激波剖面的数值格式线性化格林函数的精确描述。本文还指出了将此非线性轨道稳定性结果推广至守恒律方程组情形下离散激波剖面的可能思路。