ENERGY CONSERVING LOCAL DISCONTINUOUS GALERKIN METHODS FOR WAVE PROPAGATION PROBLEMS

Abstract. Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numer-ical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinu-ous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical exper-iments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property. 1. Introduction. Wave propagation problems arise in science, engineering and in-dustry, and they are significant to geoscience, petroleum engineering, telecommuni-cation, and the defense industry (see [14, 21] and the references therein). Among al