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

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

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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