# American Institute of Mathematical Sciences

August  2013, 7(3): 967-986. doi: 10.3934/ipi.2013.7.967

## Energy conserving local discontinuous Galerkin methods for wave propagation problems

 1 Computer Science and Mathematics Division, Oak Ridge National Laboratory, and Department of Mathematics, University of Tennessee, Oak Ridge, TN 37831, United States 2 Department of Mathematics, Mathematica Biosciences Institute, The Ohio State University, Columbus, OH 43221 3 Division of Applied Mathematics, Brown University, Providence, RI 02912

Received  May 2012 Revised  February 2013 Published  September 2013

Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous 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 experiments 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.
Citation: Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967
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