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Journal of Scientific Computing

, Volume 75, Issue 2, pp 1057–1078 | Cite as

Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

  • Marcel Klinge
  • Rüdiger Weiner
Article

Abstract

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.

Keywords

Discontinuous Galerkin Strong stability preserving Explicit peer methods 

Mathematics Subject Classification

65L05 65L06 

Notes

Acknowledgements

The authors are grateful to the anonymous referees for their valuable remarks and comments on the paper.

References

  1. 1.
    Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calvo, M., Montijano, J.I., Rández, L., Van Daele, M.: On the derivation of explicit two-step peer methods. Appl. Numer. Math. 61(4), 395–409 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186), 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cockburn, B., Shu, C.W.: The Runge-Kutta local projection \(P^1\)-discontinuous Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25(3), 337–361 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantinescu, E., Sandu, A.: Optimal explicit strong-stability-preserving general linear methods. SIAM J. Sci. Comput. 32, 3130–3150 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Horváth, Z., Podhaisky, H., Weiner, R.: Strong stability preserving explicit peer methods. In: Report 04, Martin Luther University Halle-Wittenberg, http://www.mathematik.uni-halle.de/institut/reports/ (2014)
  10. 10.
    Horváth, Z., Podhaisky, H., Weiner, R.: Strong stability preserving explicit peer methods. J. Comput. Appl. Math. 296, 776–788 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Izzo, G., Jackiewicz, Z.: Strong stability preserving general linear methods. J. Sci. Comput. 65, 271–298 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Chichester (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ketcheson, D.I., Ahmadia, A.: Optimal stability polynomials for numerical integration of initial value problems. Commun. Appl. Math. Comput. Sci. 7(2), 247–271 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge–Kutta methods. SIAM J. Numer. Anal. 49(6), 2618–2639 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Klinge, M.: Peer-Methoden für DG-Diskretisierungen. In: Master’s thesis, Martin Luther University Halle-Wittenberg (2016)Google Scholar
  16. 16.
    Kubatko, E.J., Bunya, S., Dawson, C., Westerink, J.J.: Dynamic \(p\)-adaptive Runge–Kutta discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Engrg. 198(21–26), 1766–1774 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kubatko, E.J., Westerink, J.J., Dawson, C.: Semi discrete discontinuous Galerkin methods and stage-exceeding-order strong-stability-preserving Runge–Kutta time discretizations. J. Comput. Phys. 222(2), 832–848 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kubatko, E.J., Yeager, B.A., Ketcheson, D.I.: Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods. J. Sci. Comput. 60(2), 313–344 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kulikov, G.Y., Weiner, R.: Variable-stepsize interpolating explicit parallel peer methods with inherent global error control. SIAM J. Sci. Comput. 32(4), 1695–1723 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, G., Xing, Y.: Well-balanced discontinuous Galerkin methods for the Euler equations under gravitational fields. J. Sci. Comput. 67(2), 493–513 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mirabito, C., Dawson, C., Kubatko, E.J., Westerink, J.J., Bunya, S.: Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes. Comput. Methods Appl. Mech. Eng. 200(1–4), 189–207 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. In: Technical Report LA-UR-73-479 p. Los Alamos Scientific Laboratory (1973)Google Scholar
  23. 23.
    Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 227(3), 1887–1922 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge–Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50(4), 405–418 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schmitt, B.A., Weiner, R., Beck, S.: Two-step peer methods with continuous output. BIT Numer. Math. 53(3), 717–739 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Trahan, C.J., Dawson, C.: Local time-stepping in Runge–Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations. Comput. Methods Appl. Mech. Eng. 217–220, 139–152 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weiner, R., Biermann, K., Schmitt, B.A., Podhaisky, H.: Explicit two-step peer methods. Comput. Math. Appl. 55(4), 609–619 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Weiner, R., Schmitt, B.A., Podhaisky, H., Jebens, S.: Superconvergent explicit two-step peer methods. J. Comput. Appl. Math. 223, 753–764 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of MathematicsMartin Luther University Halle-WittenbergHalle (Saale)Germany

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