A discontinuous Galerkin time integration scheme for second order differential equations with applications to seismic wave propagation problems

In this work, we present a high-order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the original equation as a system of first-order differential equations we introduce the method and show that the resulting discrete formulation is well-posed, stable, and retains a super-optimal rate of convergence with respect to the discretization parameters, namely the time step and the polynomial approximation degree. A set of two- and three-dimensional numerical experiments confirm the theoretical bounds. Finally, the method is applied to real geophysical simulations.