We discuss the rigorous justification of the spatial discretization by means of Fourier spectral methods of quasilinear first-order hyperbolic systems. We provide uniform stability estimates that grant spectral convergence of the (spatially) semi-discretized solutions towards the corresponding continuous solution provided that the underlying system satisfies some suitable structural assumptions. We consider a setting with sharp low-pass filters and a setting with smooth low-pass filters and argue that - at least theoretically - smooth low-pass filters are operable on a larger class of systems. While our theoretical results are supported with numerical evidence, we also pinpoint some behavior of the numerical method that currently has no theoretical explanation.

翻译：本文探讨了拟线性一阶双曲系统采用傅里叶谱方法进行空间离散化的严格理论依据。在系统满足适当结构假设的前提下，我们给出了保证（空间）半离散化解向对应连续解谱收敛的一致稳定性估计。研究涵盖了锐截止低通滤波与平滑低通滤波两种设置，并论证了平滑低通滤波至少在理论上适用于更广泛的系统类别。尽管数值实验支持理论结果，我们也指出了该方法当前缺乏理论解释的若干数值行为特征。