  We present and analyse a space-time discontinuous Galerkin method for wave
propagation problems. The special feature of the scheme is that it is a Trefftz
method, namely that trial and test functions are solution of the partial
differential equation to be discretised in each element of the (space-time)
mesh. The method considered is a modification of the discontinuous Galerkin
schemes of Kretzschmar et al., and of Monk and Richter. For Maxwell's equations
in one space dimension, we prove stability of the method, quasi-optimality,
best approximation estimates for polynomial Trefftz spaces and (fully explicit)
error bounds with high order in the meshwidth and in the polynomial degree. The
analysis framework also applies to scalar wave problems and Maxwell's equations
in higher space dimensions. Some numerical experiments demonstrate the
theoretical results proved and the faster convergence compared to the
non-Trefftz version of the scheme.
