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

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. (IMA J Numer Anal 36:1599–1635, 2016). 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.