SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

doi:10.4208/jcm.2003-m2018-0305

Journal of Computational Mathematics ›› 2021, Vol. 39 ›› Issue (4): 518-537.DOI: 10.4208/jcm.2003-m2018-0305

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SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

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:2019-12-21 Revised:2020-02-19 Online:2021-07-15 Published:2021-08-06
Contact: Chi-Wang Shu,Email:chi-wang shu@brown.edu
Supported by:
Research of the first author supported by the China Scholarship Council; Research of the second author supported by NSF grant DMS-1719410; Research of the third author supported by NSFC grant 11871448.

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Yong Liu, Chi-Wang Shu, Mengping Zhang. SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS[J]. Journal of Computational Mathematics, 2021, 39(4): 518-537.

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 L²-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.

Key words: Discontinuous Galerkin method, Central flux, Sub-optimal convergence rates

CLC Number:

65M60
65M15

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SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

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