Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

TL;DR

This paper introduces a novel Fourier continuation-based basis for Discontinuous Galerkin methods, enhancing the accuracy and stability of solutions for linear hyperbolic PDEs.

Contribution

It combines Fourier continuation techniques with DG methods to improve spectral accuracy and stability in solving linear hyperbolic problems.

Findings

Demonstrates high-order convergence in numerical experiments

Achieves spectrally accurate dispersion relations

Provides a stable DG framework with Fourier continuation basis

Abstract

Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. These methods have been used in PDE-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving stability and convergence. Here we propose the use of Fourier continuation in forming a new basis for the DG framework.