On the multigrid acceleration approach in computational fluid dynamics

  • Karl Solchenbach
  • Ulrich Trottenberg
Part IV - Algorithms And Applications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 295)


In this short note, two multigrid approaches for the treatment of computational fluid dynamics problems are distinguished: the “optimal approach”, where the specific model is to be treated entirely by multigrid and all multigrid components are to be defined optimally tailored - versus the “acceleration approach”, where one only tries to introduce some standard multigrid components into classical methods or into codes that are already available. For some examples, in particular the anisotropic convection-diffusion model operator and the (incompressible) Navier-Stokes equations, the gain that can be achieved by the acceleration approach is discussed.

With respect to multigrid literature, we generally refer to the multigrid bibliography


Computational Fluid Dynamic Multigrid Method Work Unit Convergence Factor Single Grid 
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.


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4. References

  1. [1]
    Barcus, M. Berechnung zweidimensionaler Strömungsprobleme mit Mehrgitterverfahren. Diplomarbeit Universität Erlangen-Nürnberg, 1987.Google Scholar
  2. [2]
    Brand, K., Lemke, M., Linden, J. Multigrid bibliography. Arbeitspapier der GMD 206, Gesellschaft für Mathematik und Datenverarbeitung, Bonn 1986.Google Scholar
  3. [3]
    Brandt, A. Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studie 85, Gesellschaft für Mathematik und Datenverarbeitung, Bonn 1984.Google Scholar
  4. [4]
    Brandt, A., Dendy, J.E., Ruppel, H. The multigrid method for semi-implicit hydrodynamics codes. J. Comp. Phys. 34, 348–370, 1980.Google Scholar
  5. [5]
    Brandt, A., Dinar, N. Multigrid solutions to ellitpic flow problems. In: Numerical Methods for Partial Differential Equations (S.V. Parker ed.), 53–147, Academic Press, New York, 1979.Google Scholar
  6. [6]
    Brockmeier, U., Mitra, N.K., Fiebig, M. Implementation of multigrid in SOLA. Notes on Numerical Fluid Mechanics, 13, 23–30, Vieweg 1986.Google Scholar
  7. [7]
    Fuchs, L. Multi-grid schemes for incompressible flows. Notes on Numerical Fluid Mechanics, 10, 38–51, Vieweg 1984.Google Scholar
  8. [8]
    Hackbusch, W. Multi-grid methods and applications. Springer Series in Computational Mathematics, 4, Springer 1985.Google Scholar
  9. [9]
    Hirt, C.W., Nichols, B.D., Romero, N.C. SOLA-a numerical solution algorithm for transient fluid flows. LASL report LA-5652, 1975.Google Scholar
  10. [10]
    Linden, J. Mehrgitterverfahren für das erste Randwertproblem der biharmonischen Gleichung und Anwendung auf ein inkompressibles Strömungsproblem. GMD-Bericht 164, Oldenbourg 1986.Google Scholar
  11. [11]
    Linden, J., Trottenberg, U., Witsch, K. Multigrid computation of the pressure of an incompressible fluid in a rotating spherical gap. Notes on Numerical Fluid Mechanics, 5, 183–193, Vieweg 1982.Google Scholar
  12. [12]
    Lonsdale, G. Solution of a rotating Navier-Stokes problem by a nonlinear multigrid algorithm. Numerical Analysis Report 105, University of Manchester, 1985.Google Scholar
  13. [13]
    Mulder, W.A. Analysis of a multigrid method for the Euler equations of gas dynamics in two dimensions. Proceedings of the 3rd International Copper Mountain Conference on Multigrid Methods, 1987.Google Scholar
  14. [14]
    Patankar, S.V., Spalding, D.B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer, 15, 1787–1806, 1972.Google Scholar
  15. [15]
    Ruttmann, B., Solchenbach, K. A multigrid solver for the computation of in-cylinder turbulent flows in engines. Notes on Numerical Fluid Mechanics, 10, 87–108, Vieweg 1984.Google Scholar
  16. [16]
    Schröder, W., Hänel, D. A comparison of several MG-methods for the solution of the time-dependent Navier-Stokes equations. In: Multigrid Methods II (W. Hackbusch, U. Trottenberg eds.), Lecture Notes in Mathematics 1228, Springer 1986.Google Scholar
  17. [17]
    Solchenbach, K., Steckel, B. Numerical simulation of the flow in 3D-cylindrical combustion chambers using multigrid methods. Arbeitspapiere der GMD 216, Gesellschaft für Mathematik und Datenverarbeitung, Bonn 1986.Google Scholar
  18. [18]
    Stone, H.L. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Num. Anal., 5, 530–560, 1968.Google Scholar
  19. [19]
    Stüben, K., Trottenberg, U. Multigrid methods: Fundamental algorithms, model problem analysis and applications. In: Multigrid Methods (W. Hackbusch, U. Trottenberg eds.), Lecture Notes in Mathematics 960, Springer 1982.Google Scholar
  20. [20]
    Thole, C.A., Trottenberg, U. Basic smoothing procedures for the multigrid treatment of elliptic 3D-operators. Notes on Numerical Fluid Mechanics, 11, 102–111, Vieweg 1984.Google Scholar
  21. [21]
    Vanka, S.P. Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comp. Phys., 65, 138–158, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Karl Solchenbach
    • 1
    • 2
  • Ulrich Trottenberg
    • 1
    • 2
    • 3
  1. 1.Suprenum GmbHBonn
  2. 2.Gesellschaft für Mathematik und Datenverarbeitung mbHSt. Augustin
  3. 3.University of CologneGermany

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