An energy-based discontinuous Galerkin method for the wave equation with advection

Abstract

An energy-based discontinuous Galerkin method for the advective wave equation is proposed and analyzed. Energy-conserving or energy-dissipating methods follow from simple, mesh-independent choices of the inter-element fluxes, and both subsonic and supersonic advection is allowed. Error estimates in the energy norm are established, and numerical experiments on structured grids display optimal convergence in the $L^2$ norm for upwind fluxes. The method generalizes earlier work on energy-based discontinuous Galerkin methods for second order wave equations which was restricted to energy forms written as a simple sum of kinetic and potential energy.