Discontinuous Galerkin discretization in time of systems of second-order nonlinear hyperbolic equations

TL;DR

This paper develops a finite element method combining discontinuous Galerkin in time and conforming finite elements in space for second-order nonlinear hyperbolic systems, providing error bounds and numerical validation.

Contribution

It introduces a novel hp-version discontinuous Galerkin method in time for nonlinear hyperbolic systems with proven error bounds and numerical verification.

Findings

Error bounds at temporal nodal points established

Numerical experiments confirm theoretical error estimates

Method effectively handles nonlinear hyperbolic equations

Abstract

In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time direction with an $H^1(\Omega)$-conforming finite element approximation in the spatial variables. Error bounds at the temporal nodal points are derived under a weak restriction on the temporal step size in terms of the spatial mesh size. Numerical experiments are presented to verify the theoretical results.