Global space-time Trefftz DG schemes for the time-dependent linear wave equation

TL;DR

This paper develops and analyzes global space-time Trefftz discontinuous Galerkin schemes for solving the time-dependent linear wave equation in anisotropic and heterogeneous media, providing optimal error estimates and high-accuracy solutions.

Contribution

It introduces new Trefftz DG methods with novel basis functions, rigorous error analysis, and strategies for handling anisotropic and heterogeneous media.

Findings

Optimal-order error estimates proved for the methods.

Numerical results confirm high accuracy of the schemes.

Effective handling of boundary conditions and media heterogeneity.

Abstract

In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth $h$ and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.