Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T02:27:24.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

Published online by Cambridge University Press:  24 May 2016

X. He  and
C. Vuik
X. He*
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, the Netherlands
C. Vuik*
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, the Netherlands
*
*Corresponding author. Email addresses: [email protected] (X. He), [email protected] (C. Vuik)
*Corresponding author. Email addresses: [email protected] (X. He), [email protected] (C. Vuik)
Get access

Abstract

In this paper we explore the performance of the SIMPLER, augmented Lagrangian, ‘grad-div’ preconditioners and their new variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem the Reynolds number varies from a low to the limiting number for a laminar flow.

Keywords

65F1065F08incompressible Navier-Stokes equationstwo-by-two block systemspreconditioners
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 239 - 261
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Axelsson, O.. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.Google Scholar
[2]Axelsson, O. and Blaheta, R.. Preconditioning of matrices partitioned in two-by-two block form: Eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Lin. Alg. Appl., 17:787–810, 2010.Google Scholar
[3]Axelsson, O. and Neytcheva, M.. A general approach to analyse preconditioners for two-by-two block matrices. Numer. Lin. Alg. Appl., article first published online: 14 DEC 2011, dpi: 10.1002/nla.830.CrossRefGoogle Scholar
[4]Axelsson, O. and Neytcheva, M.. Eigenvalue estimates for preconditioned saddle point matrices. Numer. Lin. Alg. Appl., 13:339–360, 2006.Google Scholar
[5]Benzi, M.. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys., 182:418–477, 2002.Google Scholar
[6]Benzi, M. and Olshanskii, M.A.. An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput., 28:2095–2113, 2006.CrossRefGoogle Scholar
[7]Benzi, M., Olshanskii, M.A. and Wang, Z.. Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 66:486–508, 2011.Google Scholar
[8]Benzi, M., Golub, G.H. and Liesen, J.. Numerical solution of saddle point problems. Acta Numerica, 14:1–137, 2005.Google Scholar
[9]Börm, S. and Le Borne, S.. factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems. Int. J. Numer. Meth. Fluids, 68:83–98, 2012.Google Scholar
[10]de Niet, A. and Wubs, F.W.. Two preconditioners for saddle point problems in fluid flows. Int. J. Numer. Meth. Fluids, 54:355–377, 2007.Google Scholar
[11]Eisenstat, S.C., Elman, H.C. and Schultz, M.H.. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20:345–357, 1983.CrossRefGoogle Scholar
[12]Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R. and Tuminaro, R.. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput., 27:1651–1668, 2006.CrossRefGoogle Scholar
[13]Elman, H.C. and Silvester, D.J.. Fast nonsymmetric iterations and preconditioning for the Navier-Stikes equations. SIAM J. Sci. Comput., 17:33–46, 1996.Google Scholar
[14]Elman, H.C., Silvester, D.J. and Wathen, A.J.. Finite Element and Fast Iterative Solvers: with Application in Incompressible Fluid Dynamics. Oxford Series in Numerical Mathematics and Scientific Computation, Oxford University Press: Oxford, UK, 2005.Google Scholar
[15]Elman, H.C., Silvester, D.J. and Wathen, A.J.. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math., 90:665–688, 2002.Google Scholar
[16]He, X., Neytcheva, M. and Serra Capizzano, S.. On an Augmented Lagrangian-Based Preconditioning of Oseen Type Problems. BIT Numerical Mathematics, 51:865–888, 2011.Google Scholar
[17]Heister, T.. A Massively Parallel Finite Element Framework with Application to Incompressible Flows. PhD Thesis, Göttingen University, 2011, available online.Google Scholar
[18]Kay, D., Loghin, D. and Wathen, A.. A preconditioner for the steady-state Navier-Stokes equations. SIAM J. Sci. Comput., 24:237–256, 2002.Google Scholar
[19]Klaij, C.M. and Vuik, C.. SIMPLE-type preconditioners for cell-centered, collocated finite volume discretization of incompressible Reynolds-averaged Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 71:830–849, 2013.CrossRefGoogle Scholar
[20]Li, C. and Vuik, C.. Eigenvalue analysis of the SIMPLE preconditioning for incompressible flow. Numer. Lin. Alg. Appl., 11:511–523, 2004.Google Scholar
[21]Murphy, M.F., Golub, G.H. and Wathen, A.J.. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput., 21:1969–1972, 2000.Google Scholar
[22]Napov, A. and Notay, Y.. An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput., 34:A1079–A1109, 2012.CrossRefGoogle Scholar
[23]Neytcheva, M., Do-Quang, M. and He, X.. Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form. Lecture Notes in Computer Science, 5910:108–115, 2009.Google Scholar
[24]Notay, Y.. An aggregation-based algebraic multigrid method. Electron. T. Numer. Ana., 37:123–146, 2010.Google Scholar
[25]Notay, Y.. Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput., 34:A2288–A2316, 2012.Google Scholar
[26]Notay, Y.. A new analysis of block preconditioners for saddle point problems. Report GANMN 13-01, Universite Libre de Bruxelles, Brussels, Belgium, 2013.Google Scholar
[27]Olshanskii, M.A. and Vassilevski, Y.V.. Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J. Sci. Comput., 29:2686–2704, 2007.Google Scholar
[28]Patankar, S.V.. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980.Google Scholar
[29]ur Rehman, M., Geenen, T., Vuik, C., Segal, G. and MacLachlan, S.P.. On iterative methods for the incompressible Stokes problem. Int. J. Numer. Meth. Fluids, 65:1180–1200, 2011.Google Scholar
[30]Rusten, T. and Winther, R.. A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. Appl., 13:887–904, 1992.Google Scholar
[31]Saad, Y.. Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia, PA, 2003.Google Scholar
[32]Saad, Y., van der Vorst, H.A.. Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math., 123:1–33. 2000.Google Scholar
[33]Sahin, M. and Owens, R.G.. A novel fully implicit finite volume method applied to the lid-driven cavity problem-part I: high Reynolds number flow calculations. Int. J. Numer. Meth. Fluids, 42:57–77, 2003.Google Scholar
[34]Segal, A., ur Rehman, M. and Vuik, C.. Preconditioners for the incompressible Navier-Stokes equations. Numer. Math. Theor. Meth. Appl., 3:245–275, 2010.CrossRefGoogle Scholar
[35]Vuik, C., Saghir, A. and Boerstoel, G.P.. The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces. Int. J. Numer. Meth. Fluids, 33:1027–1040, 2000.Google Scholar