Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods. (English) Zbl 0827.76069
Summary: 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.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
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. · Zbl 0641.65028 · doi:10.1137/0909009 |
| [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. · Zbl 0524.65019 · doi:10.1137/0720023 |
| [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). |
| [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. · Zbl 0718.76078 · doi:10.1007/BF02027966 |
| [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. · Zbl 0594.76023 · doi:10.1137/0907059 |
| [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. · Zbl 0505.65048 |
| [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. · Zbl 0758.76057 |
| [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. · Zbl 0761.76060 |
| [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). · Zbl 0773.76053 |
| [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. · Zbl 0773.76053 · doi:10.1006/icse.1993.1006 |
| [12] | C.W. Oosterlee and P. Wesseling, Multigrid schemes for time-dependent incompressible Navier-Stokes equations, Impact. Comp. Sci. Eng. 5 (1993) 153–175. · Zbl 0785.76058 · doi:10.1006/icse.1993.1007 |
| [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. · Zbl 0577.65086 |
| [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. · Zbl 0621.65021 |
| [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. · Zbl 0599.65018 · doi:10.1137/0907058 |
| [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. · Zbl 0753.76140 · doi:10.1002/fld.1650150404 |
| [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. · Zbl 0761.65023 · doi:10.1137/0913035 |
| [18] | H.A. van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer. Lin. Alg. Appl. 1 (1994) 369–386. · Zbl 0839.65040 · doi:10.1002/nla.1680010404 |
| [19] | C. Vuik, Further experiences with GMRESR, Supercomputer 55 (1993) 13–27. |
| [20] | C. Vuik, Solution of the discretized incompressible Navier-Stokes equations with the GMRES method, Int. J. Numer. Meth. Fluids 16 (1993) 507–523. · Zbl 0825.76552 · doi:10.1002/fld.1650160605 |
| [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. · Zbl 0844.65021 |
| [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. |
| [23] | P. Wesseling,An Introduction to Multigrid Methods (Wile, Chichester, 1992). · Zbl 0760.65092 |
| [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. · Zbl 0753.76140 |
| [25] | P.M. De Zeeuw, Matrix-dependent prolongations and restrictions in a block multigrid method solver, J. Comp. Appl. Math. 3 (1990) 1–7. · Zbl 0717.65099 |
| [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. · Zbl 0832.76063 · doi:10.1002/fld.1650200104 |
| [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. · Zbl 0822.65104 · doi:10.1137/0731090 |
| [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). · Zbl 0827.76069 |
| [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). · Zbl 0827.76069 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
