Journal of Scientific Computing

, Volume 81, Issue 3, pp 1509–1526 | Cite as

Discontinuous Galerkin Difference Methods for Symmetric Hyperbolic Systems

  • T. HagstromEmail author
  • J. W. Banks
  • B. B. Buckner
  • K. Juhnke


We develop dissipative, energy-stable difference methods for linear first-order hyperbolic systems by applying an upwind, discontinuous Galerkin construction of derivative matrices to a space of discontinuous piecewise polynomials on a structured mesh. The space is spanned by translates of a function spanning multiple cells, yielding a class of implicit difference formulas of arbitrary order. We examine the properties of the method, including the scaling of the derivative operator with method order, and demonstrate its accuracy for problems in one and two space dimensions.


Difference methods Galerkin methods Hyperbolic systems 

Mathematics Subject Classification

65M60 65M06 



  1. 1.
    Abarbanel, S., Chertock, A.: Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys. 160, 42–66 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abarbanel, S., Chertock, A., Yefet, A.: Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys. 160, 67–87 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abarbanel, S., Gottlieb, D.: A mathematical analysis of the PML method. J. Comput. Phys. 134, 357–363 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abarbanel, S., Gottlieb, D., Carpenter, M.: On the removal of boundary errors caused by Runge–Kutta integration of nonlinear partial differential equations. SIAM J. Sci. Comput. 17, 777–782 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abarbanel, S., Hesthaven, D.G.J.: Long time behavior of the perfectly matched layer equations in computational electromagnetics. J. Sci. Comput. 17, 405–422 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Abarbanel, S., Qasimov, H., Tsynkov, S.: Long-time performance of unsplit PMLs with explicit second order schemes. J. Sci. Comput. 41, 1–12 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Appelö, D., Hagstrom, T., Kreiss, G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability. SIAM J. Appl. Math. 67, 1–23 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Banks, J., Hagstrom, T.: On Galerkin difference methods. J. Comput. Phys. 313, 310–327 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Banks, J., Hagstrom, T., Jacangelo, J.: Galerkin differences for acoustic and elastic wave equations in two space dimensions. J. Comput. Phys. 372, 864–892 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Carpenter, M., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111, 220–236 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Carpenter, M., Gottlieb, D., Abarbanel, S., Don, W.S.: The theoretical accuracy of Runge–Kutta time discretizations for the initial boundary value problem. SIAM J. Sci. Comput. 16, 1241–1252 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chan, J.: Weight-adjusted discontinuous Galerkin methods: matrix-valued weights and elastic wave propagation in heterogeneous media. Int. J. Numer. Methods Eng. 113, 1779–1809 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chan, J., Hewitt, R., Warburton, T.: Weight-adjusted discontinuous Galerkin methods: curvilinear meshes. SIAM J. Sci. Comput. 39, A2395–A2421 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cockburn, B., Luskin, M., Shu, C.W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic problems. Math. Comput. 72, 577–606 (2003)CrossRefGoogle Scholar
  15. 15.
    Duru, K., Kozdon, J., Kreiss, G.: Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form. J. Comput. Phys. 303, 372–395 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Duru, K., Kreiss, G.: Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides. Wave Motion 51, 445–465 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods. John Wiley, New York (1995)zbMATHGoogle Scholar
  18. 18.
    Hagstrom, T., Hagstrom, G.: Grid stabilization of high-order one-sided differencing I: first order hyperbolic systems. J. Comput. Phys. 223, 316–340 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hagstrom, T., Hagstrom, G.: Grid stabilization of high-order one-sided differencing II: second order wave equations. J. Comput. Phys. 231, 7907–7931 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods. No. 54 in Texts in Applied Mathematics. Springer, New York (2008)CrossRefGoogle Scholar
  21. 21.
    Kozdon, J., Wilcox, L., Hagstrom, T., Banks, J.: Robust approaches to handling complex geometries with Galerkin difference methods. J. Comput. Phys. 392, 483–510 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mattsson, K., Svärd, M., Nordström, J.: Stable and accurate artificial dissipation. J. Sci. Comput. 21, 57–79 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mirzaee, H., Ryan, J., Kirby, R.: Efficient implementation of smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions. J. Sci. Comput. 52, 85–112 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Strand, B.: Summation by parts for finite difference approximations for \(d/dx\). J. Comput. Phys. 110, 47–67 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Warburton, T.: A low-storage curvilinear discontinuous Galerkin method for wave problems. SIAM J. Sci. Comput. 35, A1987–A2012 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1999)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • T. Hagstrom
    • 1
    Email author
  • J. W. Banks
    • 2
  • B. B. Buckner
    • 2
  • K. Juhnke
    • 3
  1. 1.Department of MathematicsSouthern Methodist UniversityDallasUSA
  2. 2.Rensselaer Polytechnic InstituteTroyUSA
  3. 3.Bethel CollegeNorth NewtonUSA

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