Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

doi:10.1007/s42967-019-0006-y

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (1): 101-116.doi: 10.1007/s42967-019-0006-y

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Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Yong Liu¹, Chi-Wang Shu², Mengping Zhang¹

1. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
2. Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

收稿日期:2018-06-12 修回日期:2018-11-14 出版日期:2019-03-20 发布日期:2019-05-11
通讯作者: Chi-Wang Shu, Yong Liu, Mengping Zhang E-mail:shu@dam.brown.edu;yong123@mail.ustc.edu.cn;mpzhang@ustc.edu.cn
基金资助:
C.-W.Shu:Research supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1719410;M.Zhang:Research supported by NSFC grant 11471305.

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Yong Liu¹, Chi-Wang Shu², Mengping Zhang¹

1. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
2. Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Received:2018-06-12 Revised:2018-11-14 Online:2019-03-20 Published:2019-05-11
Contact: Chi-Wang Shu, Yong Liu, Mengping Zhang E-mail:shu@dam.brown.edu;yong123@mail.ustc.edu.cn;mpzhang@ustc.edu.cn
Supported by:
C.-W.Shu:Research supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1719410;M.Zhang:Research supported by NSFC grant 11471305.

摘要/Abstract

摘要： In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in[18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be k + 2 when piecewise P^k polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise P^k polynomials with arbitrary k ≥ 1. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of k + 1 and k + 2, respectively. We also prove, under suitable choice of initial discretization, a (2k + 1)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.

关键词: Energy-conserving discontinuous Galerkin methods, Superconvergence, Linear hyperbolic equations

Abstract: In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in[18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be k + 2 when piecewise P^k polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise P^k polynomials with arbitrary k ≥ 1. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of k + 1 and k + 2, respectively. We also prove, under suitable choice of initial discretization, a (2k + 1)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.

Key words: Energy-conserving discontinuous Galerkin methods, Superconvergence, Linear hyperbolic equations

中图分类号:

65M15
65M60

Yong Liu, Chi-Wang Shu, Mengping Zhang. Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 101-116.

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Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

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