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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. 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 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.

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.

CLC Number: 

[1] J.L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comp., 82:283(2013), 1401-1432.
[2] Y. Cheng, C.S. Chou, F. Li and Y. Xing, L2 stable discontinuous Galerkin methods for onedimensional two-way wave equations, Math. Comp., 86:303(2017), 121-155.
[3] C.S. Chou, C.-W. Shu and Y. Xing, Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media, J. Comput. Phys., 272(2014), 88-107.
[4] E.T. Chung and B. Engquist, Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM J. Numer. Anal., 47:5(2009), 3820-3848.
[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, New York, 1978.
[6] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convectiondominated problems, J. Sci. Comput., 16:3(2001), 173-261.
[7] J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin methods for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics, 59:4(2019), 853-876.
[8] D.R. Durran, Numerical methods for wave equations in geophysical fluid dynamics, SpringerVerlag, New York, 1999.
[9] G. Fu and C.-W. Shu, Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems, J. Comput. Phys., 394(2019), 329-363.
[10] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev., 43:1(2001), 89-112.
[11] J. Guzmán and B. Rivière, Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations, J. Sci. Comput., 40:1-3(2009), 273-280.
[12] N.A. Kampanis, J. Ekaterinaris and V. Dougalis, Effective Computational Methods for Wave Propagation, Chapman & Hall/CRC, Boca Raton, 2008.
[13] Y. Liu, C.-W. Shu and M. Zhang, Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal., 56:1(2018), 520-541.
[14] Y. Liu, C.-W. Shu and M. Zhang, Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using P k elements, ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M 2AN), 54(2020), 705-726.
[15] X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput., 85:299(2016), 1225-1261.
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