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Optimal Superconvergence of Energy Conserving Local Discontinuous Galerkin Methods for Wave Equations

Part of: Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  05 December 2016

Waixiang Cao ,
Dongfang Li  and
Zhimin Zhang
Waixiang Cao*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China
Dongfang Li*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Zhimin Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
*
*Corresponding author. Email addresses:wxcao@csrc.ac.cn (W. Cao), dfli@hust.edu.cn (D. Li), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
*Corresponding author. Email addresses:wxcao@csrc.ac.cn (W. Cao), dfli@hust.edu.cn (D. Li), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
*Corresponding author. Email addresses:wxcao@csrc.ac.cn (W. Cao), dfli@hust.edu.cn (D. Li), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
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Abstract

This paper is concerned with numerical solutions of the LDG method for 1D wave equations. Superconvergence and energy conserving properties have been studied. We first study the superconvergence phenomenon for linear problems when alternating fluxes are used. We prove that, under some proper initial discretization, the numerical trace of the LDG approximation at nodes, as well as the cell average, converge with an order 2k+1. In addition, we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points, respectively. As a byproduct, we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp. In the second part, we propose a fully discrete numerical scheme that conserves the discrete energy. Due to the energy conserving property, after long time integration, our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.

Keywords

Local discontinuous Galerkin methods (LDG)wave equationssuperconvergenceenergy conserving

MSC classification

Secondary: 65L20: Stability and convergence of numerical methods 65M12: Stability and convergence of numerical methods 65N12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 211 - 236
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Adjerid, S. and Massey, T. C., Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 3331–3346.Google Scholar
[2] Adjerid, S. and Weinhart, T., Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3113–3129.Google Scholar
[3] Adjerid, S. and Weinhart, T., Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems, Math. Comp., 80 (2011), pp. 1335–1367.Google Scholar
[4] Anderson, M. and Kimn, J.H., A numerical approach to space-time finite elements for the wave equation, J. Comput. Phys, 226 (2007), pp. 466–476.Google Scholar
[5] Baccouch, M., A local discontinuous Galerkin method for the second-order wave equation, Comput. Meth. Appl.Mech. Eng., 212 (2012), pp. 129–143.Google Scholar
[6] Baccouch, M., Superconvergence of the local discontinuous Galerkin method applied to the one-dimensional second-order wave equation, Numer. Part. Diff. Equ., 30 (2013), pp. 862–901.Google Scholar
[7] Brunner, H., Huang, Q. and Xie, H., Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. Anal., 48 (2010), pp. 1944–1967.CrossRefGoogle Scholar
[8] Cao, W., Zhang, Z. and Zou, Q., Superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM. J. Numer. Anal., 52-5 (2014), pp. 2555–2573.Google Scholar
[9] Cao, W. and Zhang, Z., Superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Math. Comp, 85 (2016), pp. 63–84.Google Scholar
[10] Cao, W. and Zhang, Z., Point-wise and cell average error estimates for the DG and LDG method for 1D hyperbolic conservation laws and parabolic equations (in Chinese), Sci Sin Math, 45 (2015), 1115–1132.Google Scholar
[11] Cheng, Y. and Shu, C.-W., Superconvergence and time evolution of discontinuous Galerkin finite element solutions, J. Comput. Phys., 227 (2008), pp. 9612–9627.CrossRefGoogle Scholar
[12] Cheng, Y. and Shu, C.-W., Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), pp. 4044–4072.CrossRefGoogle Scholar
[13] Chou, C., Shu, C.-W. and Xing, Y., Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media, J. Comp. Phys., 272 (2014), pp. 88–107.Google Scholar
[14] Chung, E. T. and Engquist, B., Optimal discontinous Galerkin methods for wave propagation, SIAM J. Numer. Anal., 44 (2006), pp. 2131–2158.Google Scholar
[15] Chung, E. T. and Engquist, B., Optimal discontinous Galerkin methods for the acoustic wave equation in higher dimension, SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848.Google Scholar
[16] Delfour, M., Hager, W. and Trochu, F., Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36 (1981), pp. 455–473.CrossRefGoogle Scholar
[17] Duncan, D. B., Symplectic finite difference approximations of the nonlinear klein-Gordon equation, SIAM. J. Numer. Anal, 34 (1997), pp. 1742–1760.Google Scholar
[18] French, D. A. and Peterson, T. E., A continuous space-time finite element method for the wave equation, Math. Comp., 65 (1996), pp. 491–506.Google Scholar
[19] Furihata, D., Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comp. Appl. Math., 134 (2001), pp. 37–57.Google Scholar
[20] Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89–112.CrossRefGoogle Scholar
[21] Guo, W., Zhong, X. and Qiu, J., Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach, J. Comput. Phys., 235 (2013), pp. 458–485.Google Scholar
[22] Li, D. and Zhang, C., Superconvergence of a discontinuous Galerkin method for first-order linear delay differential equations, J. Comp. Math., 29 (2011), pp. 574–588.Google Scholar
[23] Li, G. and Xu, Y., Energy conserving Local discontinuous Galerkin methods for nonlinear Schrödinger equation with wave operator, J. Sci. Comput., 2014, accepted.Google Scholar
[24] Xie, Z. and Zhang, Z., Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D, Math. Comp., 79 (2010), pp. 35–45.Google Scholar
[25] Xing, Y., Chou, C.-S. and Shu, C.-W. Energy conserving discontinous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), pp. 967–986.Google Scholar
[26] Yang, Y. and Shu, C.-W., Analysis of optimal supercovergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), pp. 3110–3133.Google Scholar
[27] Yang, Y. and Shu, C.-W., Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations, J. Comp.Math., 33 (2015), pp. 323–340.Google Scholar
[28] Zhang, Z., Xie, Z. and Zhang, Z., Superconvergence of discontinuous Galerkin methods for convection-diffusion problems, J. Sci. Comput., 41 (2009), pp. 70–93.Google Scholar