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Volume 39, Issue 4
Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations

Yong Liu, Chi-Wang Shu & Mengping Zhang

DOI: 10.4208/jcm.2002-m2019-0305

J. Comp. Math., 39 (2021), pp. 518-537.

Published online: 2021-05

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  • Abstract

In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the $L^2$-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.

  • Keywords

Discontinuous Galerkin method, Central flux, Sub-optimal convergence rates.

  • AMS Subject Headings

65M60, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yong123@mail.ustc.edu.cn (Yong Liu)

Chi-Wang_Shu@brown.edu (Chi-Wang Shu)

mpzhang@ustc.edu.cn (Mengping Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-518, author = {Liu , YongShu , Chi-Wang and Zhang , Mengping}, title = {Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {4}, pages = {518--537}, abstract = {

In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the $L^2$-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2002-m2019-0305}, url = {http://global-sci.org/intro/article_detail/jcm/19158.html} }
TY - JOUR T1 - Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations AU - Liu , Yong AU - Shu , Chi-Wang AU - Zhang , Mengping JO - Journal of Computational Mathematics VL - 4 SP - 518 EP - 537 PY - 2021 DA - 2021/05 SN - 39 DO - http://doi.org/10.4208/jcm.2002-m2019-0305 UR - https://global-sci.org/intro/article_detail/jcm/19158.html KW - Discontinuous Galerkin method, Central flux, Sub-optimal convergence rates. AB -

In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the $L^2$-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.

Liu , YongShu , Chi-Wang and Zhang , Mengping. (2021). Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations. Journal of Computational Mathematics. 39 (4). 518-537. doi:10.4208/jcm.2002-m2019-0305
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