A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

TL;DR

This paper presents a novel space-time Trefftz discontinuous Galerkin method for solving the first-order acoustic wave equations, offering high-order accuracy and flexible implicit or explicit time-stepping schemes in arbitrary dimensions.

Contribution

It extends previous 1D schemes to arbitrary dimensions, introduces new Trefftz polynomial spaces, and proves optimal convergence and stability for the method.

Findings

Proves well-posedness and error bounds for the method.

Develops and analyzes two Trefftz polynomial discrete spaces.

Achieves high-order $h$-convergence in numerical experiments.

Abstract

We introduce a space-time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer. Anal., 36, 1599-1635). Test and trial discrete functions are space-time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space-time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order $h$-convergence bounds.