# 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 and Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967
##### References:
 [1] S. Adjerid and H. Temimi, A discontinuous Galerkin method for the wave equation, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 837-849. doi: 10.1016/j.cma.2010.10.008. [2] M. Baccouch, A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143. doi: 10.1016/j.cma.2011.10.012. [3] F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics, 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572. [4] Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072. doi: 10.1137/090747701. [5] E. T. Chung and B. Engquist, Optimal discontinuous Galerkin methods for wave propagation, SIAM Journal on Numerical Analysis, 44 (2006), 2131-2158. doi: 10.1137/050641193. [6] E. T. Chung and B. Engquist, Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM Journal on Numerical Analysis, 47 (2009), 3820-3848. doi: 10.1137/080729062. [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [8] B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501. [9] B. Cockburn, S.-Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One dimensional systems, Journal of Computational Physics, 84 (1989), 90-113. doi: 10.1016/0021-9991(89)90183-6. [10] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. [11] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. [12] B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, Journal of Computational Physics, 141 (1998), 199-224. doi: 10.1006/jcph.1998.5892. [13] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16 (2001), 173-261. doi: 10.1023/A:1012873910884. [14] D. R. Durran, "Numerical Methods for Wave Equations in Geophysical Fluid Dynamics," Texts in Applied Mathematics, 32, Springer-Verlag, New York, 1999. [15] R. S. Falk and G. R. Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM Journal on Numerical Analysis, 36 (1999), 935-952. doi: 10.1137/S0036142997329463. [16] L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, Mathematical Modelling and Numerical Analysis, 39 (2005), 1149-1176. doi: 10.1051/m2an:2005049. [17] D. A. French and T. E. Peterson, A continuous space-time finite element method for the wave equation, Mathematics of Computation, 65 (1996), 491-506. doi: 10.1090/S0025-5718-96-00685-0. [18] M. Grote, A. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM Journal on Numerical Analysis, 44 (2006), 2408-2431. doi: 10.1137/05063194X. [19] M. Grote and D. Schötzau, Optimal error estimates for the fully discrete interior penalty DG method for the wave equation, Journal of Scientific Computing, 40 (2009), 257-272. doi: 10.1007/s10915-008-9247-z. [20] J. S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell's equations, Journal of Computational Physics, 181 (2002), 186-221. doi: 10.1006/jcph.2002.7118. [21] N. A. Kampanis, J. Ekaterinaris and V. Dougalis, eds., "Effective Computational Methods for Wave Propagation," Numerical Insights, 5, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010879. [22] P. Monk and G. R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, Journal of Scientific Computing, 22/23 (2005), 443-477. doi: 10.1007/s10915-004-4132-5. [23] B. Rivieère and M. F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, in "Current Trends in Scientific Computing" (Xi'an, 2002), Contemporary Mathematics, 329, Amer. Math. Soc., Providence, RI, (2003), 271-282. doi: 10.1090/conm/329/05862. [24] A. Safjan and J. T. Oden, High-order taylor-galerkin and adaptive hp methods for second-order hyperbolic systems: Application to elastodynamics, Computer Methods in Applied Mechanics and Engineering, 103 (1993), 187-230. doi: 10.1016/0045-7825(93)90046-Z.

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##### References:
 [1] S. Adjerid and H. Temimi, A discontinuous Galerkin method for the wave equation, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 837-849. doi: 10.1016/j.cma.2010.10.008. [2] M. Baccouch, A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143. doi: 10.1016/j.cma.2011.10.012. [3] F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics, 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572. [4] Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072. doi: 10.1137/090747701. [5] E. T. Chung and B. Engquist, Optimal discontinuous Galerkin methods for wave propagation, SIAM Journal on Numerical Analysis, 44 (2006), 2131-2158. doi: 10.1137/050641193. [6] E. T. Chung and B. Engquist, Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM Journal on Numerical Analysis, 47 (2009), 3820-3848. doi: 10.1137/080729062. [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [8] B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501. [9] B. Cockburn, S.-Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One dimensional systems, Journal of Computational Physics, 84 (1989), 90-113. doi: 10.1016/0021-9991(89)90183-6. [10] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. [11] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. [12] B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, Journal of Computational Physics, 141 (1998), 199-224. doi: 10.1006/jcph.1998.5892. [13] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16 (2001), 173-261. doi: 10.1023/A:1012873910884. [14] D. R. Durran, "Numerical Methods for Wave Equations in Geophysical Fluid Dynamics," Texts in Applied Mathematics, 32, Springer-Verlag, New York, 1999. [15] R. S. Falk and G. R. Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM Journal on Numerical Analysis, 36 (1999), 935-952. doi: 10.1137/S0036142997329463. [16] L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, Mathematical Modelling and Numerical Analysis, 39 (2005), 1149-1176. doi: 10.1051/m2an:2005049. [17] D. A. French and T. E. Peterson, A continuous space-time finite element method for the wave equation, Mathematics of Computation, 65 (1996), 491-506. doi: 10.1090/S0025-5718-96-00685-0. [18] M. Grote, A. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM Journal on Numerical Analysis, 44 (2006), 2408-2431. doi: 10.1137/05063194X. [19] M. Grote and D. Schötzau, Optimal error estimates for the fully discrete interior penalty DG method for the wave equation, Journal of Scientific Computing, 40 (2009), 257-272. doi: 10.1007/s10915-008-9247-z. [20] J. S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell's equations, Journal of Computational Physics, 181 (2002), 186-221. doi: 10.1006/jcph.2002.7118. [21] N. A. Kampanis, J. Ekaterinaris and V. Dougalis, eds., "Effective Computational Methods for Wave Propagation," Numerical Insights, 5, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010879. [22] P. Monk and G. R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, Journal of Scientific Computing, 22/23 (2005), 443-477. doi: 10.1007/s10915-004-4132-5. [23] B. Rivieère and M. F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, in "Current Trends in Scientific Computing" (Xi'an, 2002), Contemporary Mathematics, 329, Amer. Math. Soc., Providence, RI, (2003), 271-282. doi: 10.1090/conm/329/05862. [24] A. Safjan and J. T. Oden, High-order taylor-galerkin and adaptive hp methods for second-order hyperbolic systems: Application to elastodynamics, Computer Methods in Applied Mechanics and Engineering, 103 (1993), 187-230. doi: 10.1016/0045-7825(93)90046-Z.
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