Journal of Scientific Computing

, Volume 54, Issue 1, pp 21–44 | Cite as

An Analysis of the Dissipation and Dispersion Errors of the P N P M Schemes

  • Gregor J. GassnerEmail author


We examine the dispersion and dissipation properties of the P N P M schemes for linear wave propagation problems. P N P M scheme are based on P N discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree N up to the design polynomial degree M. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (N=0) and discontinuous Galerkin schemes (N=M) are included as special cases.

We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of N and M exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.


Dispersion Dissipation Reconstructed discontinuous Galerkin PNPM schemes Wave propagation 



The author thanks the reviewers for their valuable comments. This project is kindly supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 1276: MetStroem and the research project IDIHOM within the European Research Framework Programme.


  1. 1.
    Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dumbser, M.: Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. Shaker Verlag, Aachen (2005) Google Scholar
  4. 4.
    Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.-D.: A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227, 8209–8253 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dumbser, M., Käser, M., Titarev, V.A., Toro, E.F.: Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226, 204–243 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dumbser, M., Munz, C.-D.: Arbitrary high order discontinuous Galerkin schemes. In: Cordier, S., Goudon, T., Gutnic, M., Sonnendrucker, E. (eds.) Numerical Methods for Hyperbolic and Kinetic Problems. IRMA Series in Mathematics and Theoretical Physics, pp. 295–333. EMS Publishing House, Zurich (2005) CrossRefGoogle Scholar
  7. 7.
    Dumbser, M.: Arbitrary high order pnpm schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids 39(1), 60–76 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hu, F.Q., Hussaini, M.Y., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151(2), 921–946 (1999) zbMATHCrossRefGoogle Scholar
  9. 9.
    Lo, M., van Leer, B.: Analysis and implementation of recovery-based discontinuous Galerkin for diffusion. In: 19th AIAA Computational Fluid Dynamics Conference (AIAA-2009-3786), June (2009) Google Scholar
  10. 10.
    Luo, H., Luo, L., Nourgaliev, R., Mousseau, V.A., Dinh, N.: A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids. J. Comput. Phys. 229(19), 6961–6978 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Park, H., Nourgaliev, R., Mousseau, V., Knoll, D.: Recovery discontinuous Galerkin Jacobian-free Newton Krylov method for all-speed flows. Tech. rep. inl/con-08- 13822, Idaho National Laboratory (2008) Google Scholar
  12. 12.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1999) zbMATHGoogle Scholar
  13. 13.
    van Leer, B., Lo, M.: Unification of discontinuous Galerkin methods for advection and diffusion. In: 19th AIAA Computational Fluid Dynamics Conference (AIAA-2009-0400), June (2009) Google Scholar
  14. 14.
    van Leer, B., Lo, M., van Raalte, M.: Discontinuous Galerkin method for diffusion based on recovery. In: 18th AIAA Computational Fluid Dynamics Conference (AIAA-2007-4083), June (2007) Google Scholar
  15. 15.
    van Leer, B., Nomura, S.: Discontinuous Galerkin for diffusion. In: 17th AIAA Computational Fluid Dynamics Conference (AIAA-2005-5108), 6–9 June (2005) Google Scholar
  16. 16.
    van Raalte, M., van Leer, B.: Bilinear forms for the recovery-based discontinuous Galerkin method for the diffusion. Commun. Comput. Phys. 5, 683–693 (2009) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute for Aerodynamics and GasdynamicsUniversity of StuttgartStuttgartGermany

Personalised recommendations