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：本文讨论了拟线性一阶双曲型方程组采用傅里叶谱方法进行空间离散化的严格理论论证。在底层系统满足适当结构性假设的前提下，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。我们分别考察了采用锐截止低通滤波器的设置与采用平滑低通滤波器的设置，并论证了——至少在理论上——平滑低通滤波器可在更广泛的系统类别中有效运行。虽然理论结果得到了数值证据的支持，但我们也指出了该数值方法目前尚缺理论解释的若干行为特征。