One-Dimensional Shock-Capturing for High-Order Discontinuous Galerkin Methods

  • E. Casoni
  • J. Peraire
  • A. Huerta
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 14)

Discontinuous Galerkin methods have emerged in recent years as a reasonable alternative for nonlinear conservation equations. In particular, their inherent structure (the need of a numerical flux based on a suitable approximate Riemann solver which in practice introduces some stabilization) seem to suggest that they are specially adapted to capture shocks. however, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for high-order discontinuous Galerkin. Thus, slope-limiter methods, which are extensions of finite volume methods, have been proposed for high-order approximations. Here it is shown that these techniques require mesh adaption and a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology.


Discontinuous Galerkin artificial viscosity discontinuity sensor 


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  1. 1.
    D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 1982MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131(2):267–279, 1997MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Bassi and S. Rebay. Numerical evaluation of two Discontinuous Galerkin Methods for the compressible Navier-Stokes equations. Int. J. Numer. Meth. Eng., 40, 2001Google Scholar
  4. 4.
    C. E. Baumann and J. T. Oden. An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. Int. J. Numer. Meth. Eng., 47(1–3):61–73, 2000MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Biswas, K. D. Devine, and J. E. Flaherty. Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14(1–3):255–283, 1994MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Burbeau, P. Sagaut, and C.-H. Bruneau. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys., 169(1):111–150, 2001MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    B. Cockburn. Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. J. Comput. Appl. Math., 128(1–2):187–204, 2001. Numerical analysis 2000, Vol. VII, Partial differential equationsMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    B. Cockburn, S. Y. Lin, and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys., 84(1):90–113, 1989MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp., 52(186):411– 435, 1989MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    B. Cockburn and C.-W. Shu. The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35(6):2440–2463 (electronic), 1998MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16(3):173–261, 2001MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Donea and A. Huerta. Finite element methods for flow problems. Wiley, Chichester, 2003Google Scholar
  13. 13.
    S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43(1):89–112 (electronic), 2001MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    L. Krivodonova. Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys., 226(1):879–896, 2007MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. J. LeVeque. Numerical methods for conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 1992Google Scholar
  16. 16.
    R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002MATHGoogle Scholar
  17. 17.
    S. Osher. Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal., 21(2):217–235, 1984MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. Peraire and P.-O. Persson. The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput., 30(4):1806–1824, 1988CrossRefMathSciNetGoogle Scholar
  19. 19.
    P. Persson and J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. In Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit. AIAA-2006-0112Google Scholar
  20. 20.
    J. Qiu and C.-W. Shu. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., 26(3):907–929 (electronic), 2005MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    C.-W. Shu. Total-variation-diminishing time discretizations. SIAM J. Sci. Statist. Comput., 9(6):1073–1084, 1988MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    C.-W. Shu and S. Osher. Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys., 77(2):439–471, 1988MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. Von Neumann and R. D. Richtmyer. A method for the numerical calculation of hydrody-namic shocks. J. Appl. Phys., 21:232–237, 1950MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science + Business Media B.V 2009

Authors and Affiliations

  1. 1.Laboratori de Càlcul Numèric (LaCàN), Departament de Matemàtica Aplicada, E.T.S. de Inge-nieros de Caminos, Canales y PuertosUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of Aeronautics and AstronauticsMassachusetts Institute of TechnologyUSA

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