An energy-based discontinuous Galerkin method with tame CFL numbers for the wave equation

Abstract

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers \(c \frac{\Delta t}{h} < C\) independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.