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

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.

34 pages

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• Mapped tent pitching schemes for hyperbolic systems
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• Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems
• Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation
• A space-time Trefftz discontinuous Galerkin method for the linear Schrödinger equation