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

TL;DR

This paper introduces an energy-based discontinuous Galerkin method tailored for the advective wave equation, capable of handling subsonic and supersonic flows, with proven error estimates and optimal convergence on structured grids.

Contribution

It extends energy-based DG methods to include advection, providing a flexible, mesh-independent flux choice and comprehensive error analysis for the advective wave equation.

Findings

Energy-conserving and dissipating schemes derived from flux choices.

Optimal convergence observed in numerical experiments.

Method generalizes previous energy-based DG approaches for wave equations.

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.