Abstract
The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier–Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier–Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid’s properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.
References
[1] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini, and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc, 2015.Search in Google Scholar
[2] H.J. Bandringa, High quality simulations for complex flows in maritime applications, Ph.D. thesis, Rijksuniversiteit Groningen, The Netherlands, 2016.Search in Google Scholar
[3] M. Benzi, G.H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta numerica14 (2005), 1–137.10.1017/S0962492904000212Search in Google Scholar
[4] M. Benzi and M.A. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comp. 28 (2006), 2095–2113.10.1137/050646421Search in Google Scholar
[5] M. Benzi, M.A. Olshanskii, and Z. Wang, Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations, Int. J. Numer. Methods Fluids66 (2011), 486–508.10.1002/fld.2267Search in Google Scholar
[6] M. Benzi and Z. Wang, Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations, SIAM J. Sci. Comp. 33 (2011), 2761–2784.10.1137/100797989Search in Google Scholar
[7] M. Benzi and Z. Wang, A parallel implementation of the modified augmented Lagrangian preconditioner for the incompressible Navier–Stokes equations, Numer. Alg. 64 (2013), 73–84.10.1007/s11075-012-9655-xSearch in Google Scholar
[8] J. Brown, M.G. Knepley, D.A. May, L.C. McInnes, and B. Smith, Composable linear solvers for multiphysics, In: 11th International Symposium on Parallel and Distributed Computing (ISPDC), IEEE, pp. 55–62, 2012.10.1109/ISPDC.2012.16Search in Google Scholar
[9] E.C. Cyr, J.N. Shadid, and R.S. Tuminaro, Stabilization and scalable block preconditioning for the Navier–Stokes equations, J. Comp. Phys. 231 (2012), 345–363.10.1016/j.jcp.2011.09.001Search in Google Scholar
[10] H. Elman, V.E. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro, Block preconditioners based on approximate commutators, SIAM J. Sci. Comp. 27 (2006), 1651–1668.10.1137/040608817Search in Google Scholar
[11] H. Elman, D. Silvester, and A. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2014.10.1093/acprof:oso/9780199678792.001.0001Search in Google Scholar
[12] J.H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer Science & Business Media, 2012.Search in Google Scholar
[13] D. Kay, D. Loghin, and A. Wathen, A preconditioner for the steady-state Navier–Stokes equations, SIAM J. Sci. Comp. 24 (2002), 237–256.10.1137/S106482759935808XSearch in Google Scholar
[14] C.M. Klaij, On the stabilization of finite volume methods with co-located variables for incompressible flow, J. Comp. Phys. 297 (2015), 84–89.10.1016/j.jcp.2015.05.012Search in Google Scholar
[15] C.M. Klaij and C. Vuik, SIMPLE-type preconditioners for cell-centered, colocated finite volume discretization of incompressible Reynolds-averaged Navier–Stokes equations, Int. J. Numer. Methods Fluids71 (2013), 830–849.10.1002/fld.3686Search in Google Scholar
[16] C. Li and C. Vuik, Eigenvalue analysis of the SIMPLE preconditioning for incompressible flow, Numer. Lin. Alg. Appl. 11 (2004), 511–523.10.1002/nla.358Search in Google Scholar
[17] Maritime Research Institute Netherlands, ReFRESCO Web page, http://www.refresco.org, 2015.Search in Google Scholar
[18] B.S. Massey, Units, dimensional analysis and physical similarity, Van Nostrand Reinhold company, 1971.Search in Google Scholar
[19] T.F. Miller and F.W. Schmidt, Use of a pressure-weighted interpolation method for the solution of the incompressible Navier–Stokes equations on a nonstaggered grid system, Numer. Heat Transfer, Part A: Applications14 (1988), 213–233.10.1080/10407788808913641Search in Google Scholar
[20] Numeca, Hexpress Web page, http://www.numeca.com/en/products/automeshtm/hexpresstm, 2015.Search in Google Scholar
[21] J. Pestana and A.J. Wathen, Natural preconditioning and iterative methods for saddle-point systems, SIAM Rev. 57 (2015), 71–91.10.1137/130934921Search in Google Scholar
[22] Y. Saad and Henk A. van der Vorst, Iterative solution of linear systems in the 20th century, J. Comp. Appl. Math. 123 (2000), 1–33.10.1016/B978-0-444-50617-7.50009-4Search in Google Scholar
[23] A. Segal, M. ur Rehman, and C. Vuik, Preconditioners for incompressible Navier–Stokes solvers, Numer. Math. Theor. Meth. Appl. 3 (2010), 245–275.10.4208/nmtma.2010.33.1Search in Google Scholar
[24] D. Silvester, H. Elman, D. Kay, and A. Wathen, Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow, J. Comp. Appl. Math. 128 (2001), 261–279.10.1016/B978-0-444-50616-0.50011-7Search in Google Scholar
[25] G. Vaz, F. Jaouen, and M. Hoekstra, Free-Surface Viscous Flow Computations: Validation of URANS Code FreSCo, In: 28th International Conference on Ocean, Offshore and Arctic Engineering, ASME, pp. 425–437, 2009.10.1115/OMAE2009-79398Search in Google Scholar
[26] C. Vuik, A. Saghir, and G.P. Boerstoel, The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces, Int. J. Numer. Methods Fluids33 (2000), 1027–1040.10.1002/1097-0363(20000815)33:7<1027::AID-FLD41>3.0.CO;2-SSearch in Google Scholar
[27] F.M. White, Fluid Mechanics, McGraw-Hill, New York, 1994.Search in Google Scholar
© 2017 by Walter de Gruyter Berlin/Boston
