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

2025/07/01 by Duchêne, Vincent, Marstrander, Johanna Ulvedal
#65M12 #65M70 #76M22 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2507.00516

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.

Citations

• Local well-posedness result for a class of non-local quasi-linear systems and its application to the justification of Whitham-Boussinesq systems
• Long time well-posedness of Whitham-Boussinesq systems
• Many Models for Water Waves
• Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameter
• Time integrators for dispersive equations in the long wave regime
• On Whitham and related equations
• Stability and spectral convergence of Fourier method for nonlinear\n problems. On the shortcomings of the 2/3 de-aliasing method
• On the Elimination of Aliasing in Finite-Difference Schemes by Filtering High-Wavenumber Components

Related

• A practical spectral method for hyperbolic conservation laws
• Stability and spectral convergence of Fourier method for nonlinear\n problems. On the shortcomings of the 2/3 de-aliasing method
• Classical Bounded and Almost Periodic Solutions to Quasilinear First-Order Hyperbolic Systems in a Strip
• Approximating Analytic Spectra of Hyperbolic Systems with Summation-by-Parts Finite Difference Operators
• Invariant domain preserving high-order spectral discontinuous approximations of hyperbolic systems