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Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Published online by Cambridge University Press:  03 June 2015

Mingchao Cai*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Andy Nonaka*
Affiliation:
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
John B. Bell*
Affiliation:
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Boyce E. Griffith*
Affiliation:
Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine, NY, USA
Aleksandar Donev*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
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Abstract

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well; established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Welcome, M. L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, Journal of Computational Physics, 142 (1998), pp. 146.CrossRefGoogle Scholar
[2]Almgren, A. S., Bell, J. B., and Szymczak, W. G., A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM Journal on Scientific Computing, 17 (1996), pp. 358369.Google Scholar
[3]Bell, J. B., Colella, P., and Glaz, H. M., A second order projection method for the incompressible Navier-Stokes equations, Journal of Computational Physics, 85 (1989), pp. 257283.Google Scholar
[4]Benzi, M., Preconditioning techniques for large linear systems: A survey, Journal of Computational Physics, 182 (2002), pp. 418477.Google Scholar
[5]Benzi, M., Golub, G. H., and Liesen, J., Numerical solution of saddle point problems, Acta Numerica, 14 (2005), pp. 1137.Google Scholar
[6]Briggs, W. L., Henson, V., and McCormick, S., A Multigrid Tutorial Society for Industrial and Applied Mathematics, Philadelphia, PA, (1987).Google Scholar
[7]Brown, D. L., Cortez, R., and Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, Journal of Computational Physics, 168 (2001), pp. 464499.Google Scholar
[8]Burstedde, C., Ghattas, O., Stadler, G., Tu, T., and Wilcox, L. C., Parallel scalable adjoint-based adaptive solution of variable-viscosity stokes flow problems, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 16911700.CrossRefGoogle Scholar
[9]Cahouet, J. and Chabard, J.-P., Some fast 3D finite element solvers for the generalized stokes problem, International Journal for Numerical Methods in Fluids, 8 (1988), pp. 869895.Google Scholar
[10] Z.-Cao, H., Constraint Schur complement preconditioners for nonsymmetric saddle point problems, Applied Numerical Mathematics, 59 (2009), pp. 151169.Google Scholar
[11]Chorin, A. J., Numerical solution of the Navier-Stokes equations, Journal of Computational Mathematics, 22 (1968), pp. 745762.Google Scholar
[12]Delong, S., Griffith, B. E., Vanden-Eijnden, E., and Donev, A., Temporal integrators for fluctuating hydrodynamics, Physical Review E, 87 (2013), p. 033302.CrossRefGoogle Scholar
[13]Donev, A., Fai, T. G., and Vanden-Eijnden, E., A reversible mesoscopic model of diffusion in liquids: From giant fluctuations to Fick's law, Journal of Statistical Mechanics: Theory and Experiment, 2014 (2014), p. P04004.Google Scholar
[14]Donev, A., Nonaka, A. J., Sun, Y., Fai, T. G., Garcia, A. L., and Bell, J. B., Low mach number fluctuating hydrodynamics of diffusively mixing fluids, Communications in Applied Mathematics and Computational Science, 9 (2014), pp. 47105.Google Scholar
[15]E, W. and Liu, J., Gauge method for viscous incompressible flows, Communications in Mathematical Sciences, 1 (2003), pp. 317332.Google Scholar
[16]Eiermann, M. and Ernst, O. G., Geometric aspects of the theory of Krylov subspace methods, Acta Numerica, 10 (2001), pp. 251312.CrossRefGoogle Scholar
[17]Elman, H., Howle, V. E., Shadid, J., Shuttleworth, R., and Tuminaro, R., Block preconditioners based on approximate commutators, SIAM Journal on Scientific Computing, 27 (2006), pp. 16511668.Google Scholar
[18]Elman, H., Howle, V. E., Shadid, J., Shuttleworth, R., and Tuminaro, R., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, Journal of Computational Physics, 227 (2008), pp. 17901808.Google Scholar
[19]Elman, H. C., Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM Journal on Scientific Computing, 20 (1999), pp. 12991316.Google Scholar
[20]Elman, H. C., Silvester, D. J., and Wathen, A. J., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics: with Applications in Incompressible Fluid Dynamics, OUP Oxford, 2005.Google Scholar
[21]Feng, X. and He, Y., Modified homotopy perturbation method for solving the Stokes equations, Computers & Mathematics with Applications, 61 (2011), pp. 22622266.Google Scholar
[22]Feng, X. and Shao, L., On the generalized Sor-like methods for saddle point problems, Journal of Applied Mathematics and Informatics, 28 (2010), pp. 663677.Google Scholar
[23]Fischer, B., Ramage, A., Silvester, D. J., and Wathen, A. J., Minimum residual methods for augmented systems, BIT Numerical Mathematics, 38 (1998), pp. 527543.Google Scholar
[24]Furuichi, M., May, D. A., and Tackley, P. J., Development of a stokes flow solver robust to large viscosity jumps using a Schur complement approach with mixed precision arithmetic, Journal of Computational Physics, 230 (2011), pp. 88358851.CrossRefGoogle Scholar
[25]Geenen, T., Vuik, C., Segal, G., MacLachlan, S., et al., On iterative methods for the incompressible Stokes problem, International Journal for Numerical Methods in Fluids, 65 (2011), pp.11801200.Google Scholar
[26]Gerya, T. V., May, D. A., and Duretz, T., An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity, Geochemistry, Geo-physics, Geosystems, 14 (2013), pp. 12001225.Google Scholar
[27]Griffith, B., An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner, Journal of Computational Physics, 228 (2009), pp. 75657595.Google Scholar
[28]Griffith, B., Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), pp. 317345.Google Scholar
[29]Grinevich, P., An iterative solution method for the stokes problem with variable viscosity, Moscow University Mathematics Bulletin, 65 (2010), pp. 119122.CrossRefGoogle Scholar
[30]Grinevich, P. and Olshanskii, M., An iterative method for the Stokes-type problem with variable viscosity, SIAM Journal on Scientific Computing, 31 (2009), pp. 39593978.Google Scholar
[31]Guermond, J., Minev, P., and Shen, J., An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 60116045.Google Scholar
[32]Harlow, F. and Welch, J., Numerical calculation of time-dependent viscous incompressible flow of fluids with free surfaces, Physics of Fluids, 8 (1965), pp. 21822189.Google Scholar
[33]Hu, Q. and Zou, J., Nonlinear inexact Uzawa algorithms for linear and nonlinear saddle-point problems, SIAM Journal on Optimization, 16 (2006), pp. 798825.Google Scholar
[34]Ipsen, I. C., A note on preconditioning nonsymmetric matrices, SIAM Journal on Scientific Computing, 23 (2001), pp. 10501051.Google Scholar
[35]Kay, D., Loghin, D., and Wathen, A., A preconditioner for the steady-state Navier-Stokes equations, SIAM Journal on Scientific Computing, 24 (2002), pp. 237256.Google Scholar
[36]Kay, D. A., Gresho, P. M., Griffiths, D. F., and Silvester, D. J., Adaptive time-stepping for in-compressible flow Part II: Navier-Stokes equations, SIAM Journal on Scientific Computing, 32 (2010), pp. 111128.Google Scholar
[37]Mardal, K.-A. and Winther, R., Uniform preconditioners for the time dependent stokes problem, Numerische Mathematik, 98 (2004), pp. 305327.Google Scholar
[38] —, Preconditioning discretizations of systems of partial differential equations, Numerical Linear Algebra with Applications, 18 (2011), pp. 140.Google Scholar
[39]Murphy, M. F., Golub, G. H., and Wathen, A. J., A note on preconditioning for indefinite linear systems, SIAM Journal on Scientific Computing, 21 (2000), pp. 19691972.Google Scholar
[40]Olshanskii, M., Multigrid analysis for the time dependent stokes problem, Mathematics of Computation, 81 (2012), pp. 5779.Google Scholar
[41]Olshanskii, M. A. and Chizhonkov, E. V., On the best constant in the inf-sup-condition for elongated rectangular domains, Mathematical Notes, 67 (2000), pp. 325332.Google Scholar
[42]Olshanskii, M. A., Peters, J., and Reusken, A., Uniform preconditioners for a parameter de-pendent saddle point problem with application to generalized stokes interface equations, Numerische Mathematik, 105 (2006), pp. 159191.Google Scholar
[43]Pember, R., Howell, L., Bell, J., Colella, P., Crutchfield, W., Fiveland, W., and Jessee, J., An adap-tive projection method for unsteady, low-Mach number combustion, Combustion Science and Technology, 140 (1998), pp. 123168.Google Scholar
[44]Quarteroni, A., Saleri, F., and Veneziani, A., Factorization methods for the numerical approx-imation of Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 188 (2000), pp. 505526.Google Scholar
[45]Rendleman, C., Beckner, V., Lijewski, M., Crutchfield, W., and Bell, J., Parallelization of structured, hierarchical adaptive mesh refinement algorithms, Computing and Visualization in Science, 3 (2000), pp. 147157. spoftware available at https://ccse.lbl.gov/BoxLib.Google Scholar
[46]Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM Journal on Scientific Computing, 14 (1993), pp. 461469.Google Scholar
[47]Saad, Y. and Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7 (1986), pp. 856869.Google Scholar
[48]Shin, D. and Strikwerda, J. C., Inf-sup conditions for finite-difference approximations of the Stokes equations, Journal of the Australian Mathematical Society-Series B, 39 (1997), pp. 121134.Google Scholar
[49]Turek, S., Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computional Approach, vol. 6, Springer Verlag, 1999.Google Scholar
[50]Usabiaga, F. B., Bell, J. B., Delgado-Buscalioni, R., Donev, A., Fai, T. G., Griffith, B. E., and Peskin, C. S., Staggered schemes for fluctuating hydrodynamics, SIAM Journal of Multiscale Modeling and Simulation, 10 (2012), pp. 13691408.Google Scholar
[51]Verfurth, R., A multilevel algorithm for mixed problems, SIAM Journal on Numerical Analysis, 21 (1984), pp. 264271.Google Scholar