Journal of Scientific Computing

, Volume 30, Issue 1, pp 131–175 | Cite as

Residual Distribution Schemes on Quadrilateral Meshes

  • R. Abgrall
  • F. Marpeau


We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle


Conservation laws residual distribution schemes structured and hybrid meshes non oscillatory schemes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abgrall R. (2001). Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167:277–315CrossRefMathSciNetGoogle Scholar
  2. 2.
    Deconinck, H., Sermeus, K., and Abgrall, R. (2000). Status of multidimensional upwind residual distribution schemes and applications in aeronautics. AIAA Paper 2000-2328J.Google Scholar
  3. 3.
    Deconinck, H., Struijs, R., Bourgeois, G., and Roe, P. L. (1993). Compact advection schemes on unstructured meshes. Computational Fluid Dynamics. VKI Lecture series 1993-04.Google Scholar
  4. 4.
    Paillère, H., and Deconinck, H. (1997). Euler and Navier–Stokes Solvers using Multi-dimensional Upwind Schemes and Multigrid Acceleration, chapter Compact cell vertex convection schemes on unstructured meshes, Vieweg, Braunschweig, pp. 1–49Google Scholar
  5. 5.
    van der Weide, E., and Deconinck, H. (1996). Positive matrix distribution schemes for hyperbolic systems, with application to the Euler equations. In Computational Fluid Dynamics, 3rd ECCOMAS CFD Conference, Wiley, New York.Google Scholar
  6. 6.
    Abgrall R., and Mezine M. (2003). Construction of second-order accurate monotone and stable residual distribution schemes for unsteady problems. J. Comput. Phys. 188(1):16–55CrossRefMathSciNetGoogle Scholar
  7. 7.
    Csík A., and Deconinck H. (2002). Space time residual distribution schemes for hyperbolic conservation laws on unstructured linear finite elements. In: Baines M.J (eds). Numerical Methods for Fluid Dynamics VII, Oxford, pp. 557–564.Google Scholar
  8. 8.
    van der Weide, E., and Deconinck, H. (1995). Fluctuation splitting schemes for the Euler equations on quadrilateral grids. In Numerical methods for fluid dynamics V, Oxford, UKGoogle Scholar
  9. 9.
    De Palma, P., Pascazio, G., Rubino, D. T., and Napolitano, M. (2004). Multidimensional Upwind Cell-vertex Schemes for Quadrilaterals, ECCOMAS CFD Conference 2004 JyväskylGoogle Scholar
  10. 10.
    Chou, C.-S., and Shu, C.-W. (2006). High order Residual Distribution conservative finite difference schemes for steady states problems on non smooth meshes. J. Comput. Phys. (in press)Google Scholar
  11. 11.
    Abgrall, R., Mer, K., and Nkonga, B. (2002). A Lax–Wendroff type theorem for residual schemes. In M. Hafez and J.J Chattot, editors, Innovative methods for numerical solutions of partial differential equations, World Scientific, Singapore pp. 243–266Google Scholar
  12. 12.
    Csík Á., Ricchiuto M., and Deconinck H. (2003). A conservative formulation of the multidimensional upwind residual distribution schemes for general conservation laws. J. Comput. Phys. 179(1):286–312CrossRefGoogle Scholar
  13. 13.
    Abgrall R., and Roe P.L. (2003). Construction of very high order fluctuation scheme. J. Sci. Comput 19(1–3):3–36CrossRefMathSciNetGoogle Scholar
  14. 14.
    Deconinck H., Roe P.L., and Struijs R. (1993). A multidimensional generalisation of Roe’s difference splitter for the Euler equations. Comput. Fluids 22:215–222CrossRefMathSciNetGoogle Scholar
  15. 15.
    Paillère, H. (1995). Multi-dimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructures Grids, Ph.D. thesis, Université Libre de BruxellesGoogle Scholar
  16. 16.
    Struijs, R., Deconinck, H., and Roe, P. L. (1991). Fluctuation splitting schemes for the 2D Euler equations. Computational luid Dynamics. VKI Lecture series 1991-01Google Scholar
  17. 17.
    Abgrall R., and Mezine M. (2004). Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195(2):474–507CrossRefMathSciNetGoogle Scholar
  18. 18.
    Mezine, M. (2002). Conception de schémas distributifs pour l’aérodynamique stationnaire et instationnaire. Ph.D. thesis, Université Bordeaux 1Google Scholar
  19. 19.
    Roe P.L., and Sidilkover D. (1992). Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29(6):15–42CrossRefMathSciNetGoogle Scholar
  20. 20.
    Godlewski, E., and Raviart, P. A. (1995). Numerical Approximation of Hyperbolic Systems of Conservation Laws. Volume 118 of Applied Mathematical Sciences, Springer. BerlinGoogle Scholar
  21. 21.
    Abgrall, R. (2006). Essentially non oscillatory residual distribution schemes for hyperbolic problems. J. Comput. Phys. (in press)Google Scholar
  22. 22.
    Ciarlet P.G. (1978). The Finite Element Method for Elliptic Problems. North Holland Publishing Company, AmsterdamMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Mathématiques Appliquées de BordeauxUniversité Bordeaux 1Talence CedexFrance
  2. 2.Institut Universitaire de France and Projet ScalapplixINRIA FutURsParisFrance

Personalised recommendations