Skip to main content
SpringerLink
Log in

Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We consider numerical solution methods for the incompressible Navier-Stokes equations discretized by a finite volume method on staggered grids in general coordinates. We use Krylov subspace and multigrid methods as well as their combinations. Numerical experiments are carried out on a scalar and a vector computer. Robustness and efficiency of these methods are studied. It appears that good methods result from suitable combinations of GCR and multigrid methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C.C. Ashcraft and R.G. Grimes, On vectorizing incomplete factorization and SSOR preconditioners, SIAM J. Sci. Stat. Comp. 9 (1988) 122–151.

    Article  MATH  MathSciNet  Google Scholar 

  2. S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variable iterative methods for non-symmetric systems of linear equations, SIAM J. Numer. Anal. 20 (1983) 345–357.

    Article  MATH  MathSciNet  Google Scholar 

  3. K.J. Morgan, J. Periaux and F. Thomasset (eds.),Analysis of Laminar Flow over a Backward Facing Step, GAMM Workshop held at Bièvres (Fr.) (Vieweg, Braunschweig, 1984).

    MATH  Google Scholar 

  4. A.E. Mynett, P. Wesseling, A. Segal and C.G.M. Kassels, The ISNaS incompressible Navier-Stokes solver: invariant discretization, Appl. Sci. Res. 48 (1991) 175–191.

    Article  MATH  Google Scholar 

  5. J.J.I.M. van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Stat. Comput. 7 (1986) 870–891.

    Article  MATH  Google Scholar 

  6. R. Kettler, Analysis and comparison of relaxation schemes in robust multigrid and conjugate gradient methods, in:Multigrid Methods, eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics 960 (Springer, Berlin, 1982) pp. 502–534.

    Chapter  Google Scholar 

  7. R. Kettler, Linear multigrid methods for numerical reservoir stimulation, Ph.D. Thesis, Delft University of Technology (1987).

  8. C.W. Oosterlee and P. Wesseling, A multigrid method for an invariant formulation of the incompressible Navier-Stokes equations in general co-ordinates, Commun. Appl. Numer. Meth. 8 (1992) 721–734.

    MATH  Google Scholar 

  9. C.W. Oosterlee and P. Wesseling, A robust multigrid method for a discretization of the incompressible Navier-Stokes equations in general coordinates, in:Computational Fluid Dynamics, eds. Ch. Hirsch, J. Périaux and W. Kordulla (Elsevier, Amsterdam, 1992) pp. 101–108.

    Google Scholar 

  10. C.W. Oosterlee, Robust multigrid methods for the steady and unsteady incompressible Navier-Stokes equations in general coordinates, Ph.D. Thesis, Delft University of Technology (1993).

  11. C.W. Oosterlee and P. Wesseling, A robust multigrid method for a discretization of the incompressible Navier-Stokes equations in general coordinates, Impact. Comp. Sci. Eng. 5 (1993) 128–151.

    Article  MATH  MathSciNet  Google Scholar 

  12. C.W. Oosterlee and P. Wesseling, Multigrid schemes for time-dependent incompressible Navier-Stokes equations, Impact. Comp. Sci. Eng. 5 (1993) 153–175.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Sonneveld, P. Wesseling and P.M. de Zeeuw, Multigrid and conjugate gradient methods as convergence acceleration techniques, in:Multigrid Methods for Integral and Differential Equations, eds. D.J. Paddon and H. Holstein (Clarendon Press, Oxford, 1985) pp. 117–168.

    Google Scholar 

  14. P. Sonneveld, P. Wesseling and P.M. de Zeeuw, Multigrid and conjugate gradient acceleration of basic iterative methods, in:Numerical Methods for Fluid Dynamics II, eds. K.W. Morton and M.J. Baines (Clarendon Press, Oxford, 1986) pp. 347–368.

    Google Scholar 

  15. Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comp. 7 (1986) 856–869.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Segal, P. Wesseling, J. van Kan, C.W. Oosterlee and C.G.M. Kassels, Invariant discretization of the incompressible Navier-Stokes equations in boundary fitted co-ordinates, Int. J. Numer. Meth. Fluids 15 (1992) 411–426.

    Article  MATH  Google Scholar 

  17. H.A. van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for solution of non-symmetric linear systems, SIAM J. Sci. Stat. Comp. 13 (1992) 631–644.

    Article  MATH  Google Scholar 

  18. H.A. van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer. Lin. Alg. Appl. 1 (1994) 369–386.

    Article  MATH  Google Scholar 

  19. C. Vuik, Further experiences with GMRESR, Supercomputer 55 (1993) 13–27.

    Google Scholar 

  20. C. Vuik, Solution of the discretized incompressible Navier-Stokes equations with the GMRES method, Int. J. Numer. Meth. Fluids 16 (1993) 507–523.

    Article  MATH  Google Scholar 

  21. C. Vuik, New insights in GMRES-like methods with variable preconditioners, Report 93-10, Faculty of Technical Mathematics and Informatics, TU Delft, The Netherlands (1993), to appear in J. Comp. Appl. Math.

    Google Scholar 

  22. C. Vuik, Fast iterative solvers for the discretized incompressible Navier-Stokes equations, Report 93-98, Faculty of Technical Mathematics and Informatics, TU Delft, The Netherlands (1993), to appear in Int. J. Numer. Meth. Fluids.

    Google Scholar 

  23. P. Wesseling,An Introduction to Multigrid Methods (Wile, Chichester, 1992).

    MATH  Google Scholar 

  24. P. Wesseling, A. Segal, J. van Kan, C.W. Oosterlee and C.G.M. Kassels, Finite volume discretization of the incompressible Navier-Stokes equations in general coordinates on staggered grids, Comp. Fluid Dyn. J. 1 (1992) 27–33.

    Google Scholar 

  25. P.M. De Zeeuw, Matrix-dependent prolongations and restrictions in a block multigrid method solver, J. Comp. Appl. Math. 3 (1990) 1–7.

    Article  Google Scholar 

  26. S. Zeng and P. Wesseling, An ILU smoother for the incompressible Navier-Stokes equations in general coordinates, Int. J. Numer. Meth. Fluids 20 (1995) 59–74.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Zeng and P. Wesseling, Numerical study of a multigrid method with four smoothing methods for the incompressible Navier-Stokes equations in general coordinates, in:6th Copper Mountain Conf. on Multigrid Methods, eds. N. Duane Melson, T.A. Manteuffel and S.F. McCormick, NASA Conference Pub. 3224 (1993) pp. 691–708.

  28. S. Zeng and P. Wesseling, Multigrid solution of the incompressible Navier-Stokes equations in general coordinates, SIAM J. Num. Anal. 31 (1994) 1764–1784.

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Zeng, C. Vuik and P. Wesseling, Solution of the incompressible Navier-Stokes equations in general coordinates by Krylov subspace and multigrid methods, Report 93-64, Faculty of Technical Mathematics and Informatics, TU Delft, The Netherlands (1993).

    Google Scholar 

  30. S. Zeng, C. Vuik and P. Wesseling, Further investigation on the solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods, Report 93-93, Faculty of Technical Mathematics and Informatics, TU Delft, The Netherlands (1993).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Mathematics and Informatics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    S. Zeng, C. Vuik & P. Wesseling

Authors
  1. S. Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Vuik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Wesseling
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, S., Vuik, C. & Wesseling, P. Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods. Adv Comput Math 4, 27–49 (1995). https://doi.org/10.1007/BF02123472

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02123472

Keywords

Search

Navigation