Communications on Applied Mathematics and Computation >

2019 , Vol. 1 >Issue 1: 101 - 116

DOI: https://doi.org/10.1007/s42967-019-0006-y

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

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  • 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 date: 2018-06-12

  Revised date: 2018-11-14

  Online published: 2019-05-11

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.

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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 Pk polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise Pk 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

Cite this article

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 . DOI: 10.1007/s42967-019-0006-y

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