A high-order discontinuous Galerkin in time discretization for second-order hyperbolic equations

The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators undiscretized. The well-posedness of this semi-discrete scheme is analyzed and a priori error estimates are derived in the energy norm. We then combine this $hp$-version discontinuous Galerkin method for temporal discretization with an $H^{1$-conforming} finite element approximation for the spatial variables to construct a fully discrete scheme. A prior error estimates are derived both in the energy norm and the $L^2$-norm. Numerical experiments are presented to verify the theoretical results.