Variational Space-Time Methods for the Wave Equation

The accurate and reliable numerical approximation of the hyperbolic wave equation is of fundamental importance to the simulation of acoustic, electromagnetic and elastic wave propagation phenomena. Here, we present families of variational space–time discretisation methods of higher order for the acoustic wave equation as a protoype model for the three-dimensional elastic wave equation. In temporal domain we present two families of continuous and discontinuous discretisation schemes. For the spatial discretisation a symmetric interior penalty discontinuous Galerkin method is used. From these classes of uniform Galerkin discretisations in space and time an approach of fourth-order accuracy is analysed carefully. The efficient solution of the resulting block-matrix system and inherently parallel numerical simulation through domain decomposition is adressed. The performance properties of the schemes are illustrated by sophisticated and challenging numerical experiments with complex wave propagation phenomena in heterogeneous media.