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

  • Yong Liu
  • Chi-Wang ShuEmail author
  • Mengping Zhang
Original Paper

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 \(\mathbb {P}^k\) polynomials with \(k \ge 1\) are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise \(\mathbb {P}^k\) polynomials with arbitrary \(k \ge 1\). 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.

Keywords

Energy-conserving discontinuous Galerkin methods Superconvergence Linear hyperbolic equations 

Mathematics Subject Classification

65M15 65M60 

References

  1. 1.
    Adjerid, S., Massey, T.C.: Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Eng. 195, 3331–3346 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adjerid, S., Weinhart, T.: Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems. Comput. Methods Appl. Mech. Eng. 198, 3113–3129 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adjerid, S., Weinhart, T.: Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems. Mathe. Comput. 80, 1335–1367 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, W., Li, D., Yang, Y., Zhang, Z.: Superconvergence of discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations. ESAIM. 51, 467–486 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cao, W., Liu, H., Zhang, Z.: Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations. Numer. Methods Partial Differ. Equ. 33, 290–317 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cao, W., Shu, C.-W., Yang, Y., Zhang, Z.: Superconvergence of discontinuous Galerkin method for nonlinear hyperbolic equations. SIAM J. Numer. Anal. 56, 732–765 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cao, W., Shu, C.-W., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients. ESAIM. 51, 2213–2235 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cao, W., Zhang, Z., Zou, Q.: Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 52, 2555–2573 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227, 9612–9627 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam, New York (1978)Google Scholar
  12. 12.
    Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52, 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cockburn, B., Shu, C.-W.: The Runge–Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337–361 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Durran, D.R.: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, New York (1999)CrossRefGoogle Scholar
  18. 18.
    Fu, G., Shu, C.-W.: Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems. Submitted to Journal of Computational Physics. arXiv:1805.04471
  19. 19.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Guo, W., Zhong, X., Qiu, J.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235, 458–485 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kampanis, N.A., Ekaterinaris, J., Dougalis, V.: Effective Computational Methods for Wave Propagation. Chapman & Hall/CRC, Boca Raton (2008)zbMATHGoogle Scholar
  22. 22.
    Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323–338 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, Y., Shu, C.-W., Zhang, M.: Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 56, 520–541 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xie, Z., Zhang, Z.: Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comput. 79, 35–45 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, Z., Xie, Z., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for convection-diffusion problems. J. Sci. Comput. 41, 70–93 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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