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

TL;DR

This paper introduces a high-order discontinuous Galerkin time integration scheme for second-order differential equations, demonstrating stability, convergence, and applicability to seismic wave propagation through numerical experiments and real-world geophysical applications.

Contribution

The paper develops a novel high-order DG time integration method for second-order systems, ensuring stability and super-optimal convergence, with practical applications to seismic wave modeling.

Findings

Method is well-posed and stable

Achieves super-optimal convergence rates

Successfully applied to seismic wave problems

Abstract

In this work, we present a new 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 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 applications.