The Fourier spectral approach to the spatial discretization of quasilinear hyperbolic systems

Abstract 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) semidiscretized 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.