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

TL;DR

The paper addresses rigorously justifying Fourier spectral spatial discretization for quasilinear hyperbolic systems, proving uniform stability and spectral convergence of semi-discrete solutions under Friedrichs-symmetrizable structures. It develops and analyzes sharp and smooth low-pass filtering strategies, establishing energy estimates and convergence for both symmetric and symmetrizable systems, with sharper requirements for sharp filters in the symmetrizable case. Key contributions include precise convergence statements (e.g., rate estimates in $H^r$ norms) and the extension to Hamiltonian-inspired Saint-Venant models, complemented by numerical experiments that confirm spectral convergence and reveal nuanced stability behavior depending on filter type and hyperbolicity. The findings provide a rigorous foundation for employing Fourier spectral methods in quasilinear hyperbolic PDEs and offer practical guidance on filter choice, especially highlighting the broader applicability of smooth filters and the importance of hyperbolicity-domain considerations in the semi-discrete setting.

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.

Figures (7)

• Figure 1: Experiments with initial data \ref{['eq.init1']}.
• Figure 2: Experiments with initial data \ref{['eq.init1']}.
• Figure 3: Experiments with initial data \ref{['eq.init2']}.
• Figure 4: Experiments with initial data \ref{['eq.init_zero_depth']}.
• Figure 5: Plot illustrating the convergence of the numerical schemes \ref{['eq.SV-sharp']} and \ref{['eq.SV-smooth']} for the systems \ref{['eq.SV']} (in the left) and \ref{['eq.SV-2']} (in the right) in two spatial dimensions as the number of collocation points $2M$ increases. The plot shows the relative error of the numerical solution for initial data \ref{['eq.init2D']} with $h_0=0.5, u_{{\rm l}} =- v_{{\rm l}} = 0.5, u_{{\rm h}} = -v_{{\rm h}} = 1$ and $s=2$ at time $T=0.1$. The initial data is in in $H^2((2\pi\mathbb{T})^2)^3$ and the relative error is measured in the $L^2$-norm, $E_0$ and in the $H^1$-norm, $E_1$ for $2M = 2^j, j= 6,\ldots, 9$ when using either sharp or smooth low-pass filters. To illustrate, the blue and orange lines have slopes $-2$ and $-1$ respectively. The numerical scheme exhibits spectral convergence with both sharp and smooth low-pass filters, for both systems.

Theorems & Definitions (40)

• Remark 1.1
• Remark 1.2
• Proposition 2.1: Well-posedness
• Proposition 2.2: Uniform estimates
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop.sym_sharp_num_bound']}.
• Proposition 2.4: Convergence
• proof
• Proposition 2.5: Uniform estimates