We introduce a space-time discretization for elastic and acoustic waves using a
discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. For the
dG method, the upwind flux is evaluated by explicitly solving a Riemann problem. Then we
show well-posedness and convergence of the discrete system. Based on goal-oriented dualweighted
error estimation an adaptive strategy is introduced. The full space-time linear system
is solved with a parallel multilevel preconditioner. Numerical experiments for acoustic and
elastic waves underline the efficiency of the overall adaptive solution process.