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.
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C.C. Ashcraft and R.G. Grimes, On vectorizing incomplete factorization and SSOR preconditioners, SIAM J. Sci. Stat. Comp. 9 (1988) 122–151.
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.
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).
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.
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.
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.
R. Kettler, Linear multigrid methods for numerical reservoir stimulation, Ph.D. Thesis, Delft University of Technology (1987).
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.
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.
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).
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.
C.W. Oosterlee and P. Wesseling, Multigrid schemes for time-dependent incompressible Navier-Stokes equations, Impact. Comp. Sci. Eng. 5 (1993) 153–175.
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.
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.
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.
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.
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.
H.A. van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer. Lin. Alg. Appl. 1 (1994) 369–386.
C. Vuik, Further experiences with GMRESR, Supercomputer 55 (1993) 13–27.
C. Vuik, Solution of the discretized incompressible Navier-Stokes equations with the GMRES method, Int. J. Numer. Meth. Fluids 16 (1993) 507–523.
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.
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.
P. Wesseling,An Introduction to Multigrid Methods (Wile, Chichester, 1992).
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.
P.M. De Zeeuw, Matrix-dependent prolongations and restrictions in a block multigrid method solver, J. Comp. Appl. Math. 3 (1990) 1–7.
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.
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.
S. Zeng and P. Wesseling, Multigrid solution of the incompressible Navier-Stokes equations in general coordinates, SIAM J. Num. Anal. 31 (1994) 1764–1784.
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).
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).
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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
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DOI: https://doi.org/10.1007/BF02123472