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A new space-time discontinuous Galerkin (dG) method utilising special Trefftzpolynomial basis functions is proposed and fully analysed for the scalar waveequation in a second order formulation. The dG method considered is motivatedby the class of interior penalty dG methods, as well as by the classical workof Hughes and Hulbert. The choice of the penalty terms included in the bilinearform is essential for both the theoretical analysis and for the practicalbehaviour of the method for the case of lowest order basis functions. A bestapproximation result is proven for this new space-time dG method withTrefftz-type basis functions. Rates of convergence are proved in any dimensionand verified numerically in spatial dimensions $d = 1$ and $d = 2$. Numericalexperiments highlight the effectiveness of the Trefftz method in problems withenergy at high frequencies.

A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation