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A New hp-Adaptive DG Scheme for Conservation Laws Based on Error Control

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Hyperbolic Problems: Theory, Numerics, Applications

In this contribution we present a new hp-adaptive Discontinuous Galerkin approximation for scalar conservation laws. The hp-adaptivity of the scheme is based on a rigorous a posteriori error estimate for a generalized class of Discontinuous Galerkin schemes presented in [DMO06]. Numerical experiments demonstrate the efficiency and stability of the scheme in multiple space dimensions.

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References

  1. S. Adjerid, K.D. Devine, J.E. Flaherty, and L. Krivodonova. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 191:1097–1112, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  2. ALUGrid Library homepage. http://www.mathematik.uni-freiburg.de/IAM/Research/alugrid.

  3. A. Burri, A. Dedner, D. Diehl, R. Klöfkorn, and M. Ohlberger. A general object oriented framework for discretizing nonlinear evolution equations. In Proceedings of the 1st Kazakh-German Advanced Research Workshop on Computational Science and High Performance Computing, Almaty, Kazakhstan, September 25 - October 1, 2005.

    Google Scholar 

  4. P. Bastian, M. Droske, C. Engwer, R. Klöfkorn, T. Neubauer, M. Ohlberger, and M. Rumpf. Towards a unified framework for scientific computing. In Proceedings of the 15th International Conference on Domain Decomposition Methods, Berlin, July 21-25, 2003.

    Google Scholar 

  5. R. Biswas, K.D. Devine, and J.E. Flaherty. Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14:255–283, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Cockburn and P.A. Gremaud. Error estimates for finite element methods for scalar conservation laws. SIAM J. Num. Anal., 33:522–554, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Cockburn, S. Hou, and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comp., 54(190):545–581, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Cockburn and C.-W. Shu. The Runge-Kutta local projection P 1-discontinuous-Galerkin finite element method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér., 25(3):337–361, 1991.

    MATH  MathSciNet  Google Scholar 

  9. B. Cockburn and C.-W. Shu. Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16(3): 173–261, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Dedner, C. Makridakis, and M. Ohlberger. Error control for a class of Runge Kutta Discontinuous Galerkin methods for nonlinear conservation laws. Accepted for publication in SIAM J. Numer. Anal., 2006.

    Google Scholar 

  11. S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43(1):89–112, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Hartmann and P. Houston. Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput., 24(3):979–1004, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Houston and E. Süli. h p-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J. Sci. Comput., 23(4):1226–1252, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Houston, B. Senior, and E. Süli. h p-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity. Internat. J. Numer. Methods Fluids, 40(1-2):153–169, 2002. ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001).

    Google Scholar 

  15. J. Jaffré, C. Johnson, and A. Szepessy. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci., 5(3):367–386, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Johnson and A. Szepessy. Adaptive finite element methods for conservation laws based on a posteriori error estimates. Commun. Pure Appl. Math., 48:199–234, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  17. M.G. Larson and T.J. Barth. A posteriori error estimation for adaptive discontinuous Galerkin approximations of hyperbolic systems. In Discontinuous Galerkin methods (Newport, RI, 1999), volume 11 of Lect. Notes Comput. Sci. Eng., pages 363–368. Springer, Berlin, 2000.

    Google Scholar 

  18. E. Süli and P. Houston. Adaptive finite element approximation of hyperbolic problems. In Error estimation and adaptive discretization methods in computational fluid dynamics, volume 25 of Lect. Notes Comput. Sci. Eng., pages 269–344. Springer, Berlin, 2003.

    Google Scholar 

  19. C.-W. Shu. A survey of strong stability preserving high order time discretizations. Scientific Computing Report Series (2001-18), Brown University, 2001.

    Google Scholar 

  20. M. Zennaro. Natural continuous extensions of Runge-Kutta methods. Math. Comput., 46:119–133, 1986.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Universität Freiburg, Herrmann-Herder-Str. 10, 79104, Freiburg, Germany

    A. Dedner & M. Ohlberger

Authors
  1. A. Dedner
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  2. M. Ohlberger
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Editors and Affiliations

  1. Institut Camille Jordan UMR CNRS 5208, Université Claude Bernard Lyon 1, 43, Bd du, 69622, Villeurbanne cedex, France

    Sylvie Benzoni-Gavage

  2. UMPA, UMR CNRS 5669, Ecole Normale Supérieure de Lyon, 46, allée d'ltalie, 69364, Lyon cedex 07, France

    Denis Serre

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Dedner, A., Ohlberger, M. (2008). A New hp-Adaptive DG Scheme for Conservation Laws Based on Error Control. In: Benzoni-Gavage, S., Serre, D. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75712-2_15

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