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

TL;DR

This paper rigorously justifies the use of Fourier spectral methods for spatial discretization of quasilinear hyperbolic systems, establishing stability and convergence under certain conditions, supported by numerical evidence.

Contribution

It provides the first rigorous stability and convergence analysis for Fourier spectral discretization of quasilinear hyperbolic systems, including the effects of different low-pass filters.

Findings

Spectral convergence is achieved under structural assumptions.

Smooth low-pass filters work on a broader class of systems.

Numerical experiments reveal behaviors not yet explained theoretically.

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) 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.