Incorporating a Discontinuous Galerkin Method into the Existing Vertex-Centered Edge-Based Finite Volume Solver Edge

  • Sven-Erik Ekström
  • Martin Berggren
Conference paper
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 113)


The discontinuous Galerkin (DG) method can be viewed as a generalization to higher orders of the finite volume method. At lowest order, the standard DG method reduces to the cell-centered finite volume method.We introduce for the Euler equations an alternative DG formulation that reduces to the vertex-centered version of the finite volume method at lowest order. The method has been successfully implemented for the Euler equations in two space dimensions, allowing a local polynomial order up to p=3 and supporting curved elements at the airfoil boundary. The implementation has been done as an extension within the existing edge-based vertex-centered finite-volume code Edge.


Euler Equation Control Volume Discontinuous Galerkin Discontinuous Galerkin Method Dual Cell 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114(1), 45–58 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abgrall, R.: Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167(2), 277–315 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barth, T.J., Jespersen, D.C.: The design and application of upwind schemes on unstructured meshes. In: 27th Aerospace Sciences Meeting, AIAA 89-0366, Reno, Nevada (1989)Google Scholar
  4. 4.
    Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2d Euler equations. J. Comp. Phys. 138, 251–285 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berggren, M.: A vertex-centered, dual discontinuous Galerkin method. J. Comput. Appl. Math. 192(1), 175–181 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Berggren, M., Ekström, S.-E., Nordström, J.: A discontinuous Galerkin extension of the vertex-centered edge-based finite volume method. Commun. Comput. Phys. 5, 456–468 (2009)MathSciNetGoogle Scholar
  7. 7.
    Blazek, J.: Computational Fluid Dynamics, 2nd edn. Elsevier, Amsterdam (2005)Google Scholar
  8. 8.
    Catabriga, L., Coutinho, A.: Implicit SUPG solution of Euler equations using edge-based data structures. Comput. Meth. Appl. Mech. Eng. 191(32), 3477–3490 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Csík, A., Ricchiuto, M., Deconinck, H.: A Conservative Formulation of the Multidimensional Upwind Residual Distribution Schemes for General Nonlinear Conservation Laws. J. Comput. Phys. 179(1), 286–312 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ekström, S.-E., Berggren, M.: Agglomeration multigrid for the vertex-centered dual discontinuous Galerkin method. In: Kroll, N., et al. (eds.) ADIGMA. NNFM, vol. 113, pp. 301–308. Springer, Heidelberg (2010)Google Scholar
  12. 12.
    Eliasson, P.: EDGE, a Navier–Stokes solver, for unstructured grids. Technical Report FOI-R-0298-SE, Swedish Defence Research Agency (2001)Google Scholar
  13. 13.
    Fezoui, L., Stoufflet, B.: A class of implicit upwind schemes for Euler simulations with unstructured meshes. J. Comput. Phys. 84(1), 174–206 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    FOI. Edge - Theoretical Formulation. Technical Report FOI dnr 03-2870, Swedish Defence Research Agency (2007) ISSN-1650-1942Google Scholar
  15. 15.
    Fun3D. Fully Unstructured Navier–Stokes,
  16. 16.
    Haselbacher, A., McGuirk, J.J., Page, G.J.: Finite volume discretization aspects for viscous flows on mixed unstructured grids. AIAA J. 37(2), 177–184 (1999)CrossRefGoogle Scholar
  17. 17.
    Hirsch, C.: Numerical Computation of Internal and External Flows, vol. 2. Wiley, Chichester (1990)zbMATHGoogle Scholar
  18. 18.
    Krivodonova, L., Berger, M.: High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comp. Phys. 211(2), 492–512 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Luo, H., Baum, J.D., Löhner, R.: Edge-based finite element scheme for the Euler equations. AIAA J. 32, 1182–1190 (1994)CrossRefGoogle Scholar
  20. 20.
    Morton, K.W., Sonar, T.: Finite volume methods for hyperbolic conservation laws. Acta Numer. 16, 155–238 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Persson, P.-O., Peraire, J.: Sub-Cell shock Capturing for Discontinuous Galerkin Methods. AIAA Paper, 2006-112 (2006)Google Scholar
  22. 22.
    Schwamborn, D., Gerhold, T., Heinrich, R.: The DLR TAU-Code: Recent Applications in Research and Industry. In: Wesseling, P., Onate, E., Périaux, J. (eds.) ECCOMAS CFD 2006 (2006)Google Scholar
  23. 23.
    Smith, T.M., Ober, C.C., Lorber, A.A.: SIERRA/Premo–A New General Purpose Compressible Flow Simulation Code. In: AIAA 32nd Fluid Dynamics Conference, St. Louis (2002)Google Scholar
  24. 24.
    Solin, P., Segeth, K., Dolezel, I.: Higher Order Finite Element Methods. Taylor & Francis Ltd., Abington (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Sven-Erik Ekström
    • 1
  • Martin Berggren
    • 2
  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Department of Computing ScienceUmeå UniversityUmeåSweden

Personalised recommendations