In this article we study the spectral, linear and nonlinear stability of stationary shock profile solutions to the Lax-Wendroff scheme for hyperbolic conservation laws. We first clarify the spectral stability of such solutions depending on the convexity of the flux for the underlying conservation law. The main contribution of this article is a detailed study of the so-called Green's function for the linearized numerical scheme. As evidenced on numerical simulations, the Green's function exhibits a highly oscillating behavior ahead of the leading wave before this wave reaches the shock location. One of our main results gives a quantitative description of this behavior. Because of the existence of a one-parameter family of stationary shock profiles, the linearized numerical scheme admits the eigenvalue 1 that is embedded in its continuous spectrum, which gives rise to several contributions in the Green's function. Our detailed analysis of the Green's function describes these contributions by means of a so-called activation function. For large times, the activation function describes how the mass of the initial condition accumulates along the eigenvector associated with the eigenvalue 1 of the linearized numerical scheme. We can then obtain sharp decay estimates for the linearized numerical scheme in polynomially weighted spaces, which in turn yield a nonlinear orbital stability result for spectrally stable stationary shock profiles. This nonlinear result is obtained despite the lack of uniform ${\ell}$ 1 estimates for the Green's function of the linearized numerical scheme, the lack of such estimates being linked with the dispersive nature of the numerical scheme. This dispersive feature is in sharp contrast with previous results on the orbital stability of traveling waves or discrete shock profiles for parabolic perturbations of conservation laws.

翻译：本文研究双曲守恒律拉克斯-温德罗夫格式稳态激波剖面解的谱稳定性、线性稳定性及非线性稳定性。我们首先阐明了此类解的谱稳定性与底层守恒律通量凸性之间的依赖关系。本文主要贡献在于对线性化数值格式的格林函数进行了细致研究。数值模拟表明，格林函数在主导波前到达激波位置前会呈现高度振荡特性。我们的核心成果之一对此行为给出了定量描述。由于存在单参数族的稳态激波剖面，线性化数值格式在连续谱中嵌入了特征值1，导致格林函数中出现若干贡献项。我们通过对格林函数的精细分析，借助所谓激活函数描述了这些贡献项。在长时间尺度下，激活函数刻画了初始条件质量如何沿线性化数值格式特征值1对应的特征向量累积。基于此，我们能在多项式加权空间中获得线性化数值格式的锐利衰减估计，进而推导出谱稳定稳态激波剖面的非线性轨道稳定性结果。尽管线性化数值格式的格林函数缺乏一致的${\ell}$ 1估计（该缺陷与数值格式的色散特性相关），我们仍能获得此非线性结论。这种色散特性与先前关于守恒律抛物扰动行波或离散激波剖面轨道稳定性的研究形成鲜明对比。