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An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner. (English) Zbl 1391.76474

Summary: The projection method is a widely used fractional-step algorithm for solving the incompressible Navier-Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier-Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids

Software:

Gerris; hypre; PETSc
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Full Text: DOI

References:

[1] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 104, 745-762 (1968) · Zbl 0198.50103
[2] Chorin, A. J., On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comput., 23, 106, 341-353 (1969) · Zbl 0184.20103
[3] Minion, M. L., A projection method for locally refined grids, J. Comput. Phys., 127, 1, 158-178 (1996) · Zbl 0859.76047
[4] Almgren, A. S.; Bell, J. B.; Colella, P.; Marthaler, T., A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. Sci. Comput., 18, 5, 1289-1309 (1997) · Zbl 0910.76040
[5] Martin, D. F.; Colella, P., A cell-centered adaptive projection method for the incompressible Euler equations, J. Comput. Phys., 163, 2, 271-312 (2000) · Zbl 0991.76052
[6] Almgren, A. S.; Bell, J. B.; Crutchfield, W. Y., Approximate projection methods: Part I. Inviscid analysis, SIAM J. Sci. Comput., 22, 4, 1139-1159 (2000) · Zbl 0995.76059
[7] Popinet, S., Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys., 190, 2, 572-600 (2003) · Zbl 1076.76002
[8] Kadioglu, S. Y.; Klein, R.; Minion, M. L., A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics, J. Comput. Phys., 227, 3, 2012-2043 (2008) · Zbl 1146.76035
[9] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323 (1985) · Zbl 0582.76038
[10] Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 2, 257-283 (1989) · Zbl 0681.76030
[11] Beyer, R. P., A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98, 1, 145-162 (1992) · Zbl 0744.76128
[12] Tau, E. Y., A 2nd-order projection method for the incompressible Navier-Stokes equations in arbitrary domains, J. Comput. Phys., 115, 1, 147-152 (1994) · Zbl 0811.76064
[13] Almgren, A. S.; Bell, J. B.; Szymczak, W. G., A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17, 2, 358-369 (1996) · Zbl 0845.76055
[14] Howell, L. H.; Bell, J. B., An adaptive mesh projection method for viscous incompressible flow, SIAM J. Sci. Comput., 18, 4, 996-1013 (1997) · Zbl 0901.76057
[15] Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142, 1, 1-46 (1998) · Zbl 0933.76055
[16] Rider, W. J., Filtering non-solenoidal modes in numerical solutions of incompressible flows, Int. J. Numer. Methods Fluid, 28, 5, 789-814 (1998) · Zbl 0931.76062
[17] Roma, A. M.; Peskin, C. S.; Berger, M. J., An adaptive version of the immersed boundary method, J. Comput. Phys., 153, 2, 509-534 (1999) · Zbl 0953.76069
[18] Brown, D. L.; Cortez, R.; Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 2, 464-499 (2001) · Zbl 1153.76339
[19] Li, Z.-L.; Lai, M.-C., The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171, 2, 822-842 (2001) · Zbl 1065.76568
[20] Lee, L.; LeVeque, R. J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 25, 3, 832-856 (2003) · Zbl 1163.65322
[21] Guy, R. D.; Fogelson, A. L., Stability of approximate projection methods on cell-centered grids, J. Comput. Phys., 203, 2, 517-538 (2005) · Zbl 1143.76558
[22] Griffith, B. E.; Peskin, C. S., On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems, J. Comput. Phys., 208, 1, 75-105 (2005) · Zbl 1115.76386
[23] Yang, B.; Prosperetti, A., A second-order boundary-fitted projection method for free-surface flow computations, J. Comput. Phys., 213, 2, 574-590 (2006) · Zbl 1136.76415
[24] Min, C.; Gibou, F., A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids, J. Comput. Phys., 219, 2, 912-929 (2006) · Zbl 1330.76096
[25] Zheng, Z.; Petzold, L., Runge-Kutta-Chebyshev projection method, J. Comput. Phys., 219, 2, 976-991 (2006) · Zbl 1103.76048
[26] Griffith, B. E.; Hornung, R. D.; McQueen, D. M.; Peskin, C. S., An adaptive, formally second order accurate version of the immersed boundary method, J. Comput. Phys., 223, 1, 10-49 (2007) · Zbl 1163.76041
[27] Martin, D. F.; Colella, P.; Graves, D., A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions, J. Comput. Phys., 227, 3, 1863-1886 (2008) · Zbl 1137.76040
[28] Le, D. V.; Khoo, B. C.; Lim, K. M., An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains, Comput. Methods Appl. Mech. Eng., 197, 2119-2130 (2008) · Zbl 1158.76407
[29] Griffith, B. E.; Luo, X.; McQueen, D. M.; Peskin, C. S., Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method, Int. J. Appl. Mech., 1, 1, 137-177 (2009)
[30] Guermond, J. L.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, 44-47, 6011-6045 (2006) · Zbl 1122.76072
[31] Rider, W. J.; Greenough, J. A.; Kamm, J. R., Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations, J. Comput. Phys., 225, 2, 1827-1848 (2007) · Zbl 1343.76036
[32] Colella, P.; Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 1, 174-201 (1984) · Zbl 0531.76082
[33] Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 3, 387-411 (1982) · Zbl 0511.76031
[34] Botella, O.; Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluid, 27, 4, 421-433 (1998) · Zbl 0964.76066
[35] Erturk, E.; Corke, T. C.; Gökçöl, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Methods Fluid, 48, 7, 747-774 (2005) · Zbl 1071.76038
[36] Roy, C. J.; Sinclair, A. J., On the generation of exact solutions for evaluating numerical schemes and estimating discretization error, J. Comput. Phys., 228, 5, 1790-1802 (2009) · Zbl 1159.65093
[37] Kay, D.; Loghin, D.; Wathen, A., A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24, 237-256 (2002) · Zbl 1013.65039
[38] Silvester, D.; Elman, H.; Kay, D.; Wathen, A., Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow, J. Comput. Appl. Math., 128, 1-2, 261-279 (2001) · Zbl 0983.76051
[39] Elman, H. C.; Howle, V. E.; Shadid, J. N.; Tuminaro, R. S., A parallel block multi-level preconditioner for the 3D incompressible Navier-Stokes equations, J. Comput. Phys., 187, 2, 504-523 (2003) · Zbl 1061.76058
[40] Elman, H.; Howle, V. E.; Shadid, J.; Shuttleworth, R.; Tuminaro, R., Block preconditioners based on approximate commutators, SIAM J. Sci. Comput., 27, 5, 1651-1668 (2006) · Zbl 1100.65042
[41] Elman, H.; Howle, V. E.; Shadid, J.; Shuttleworth, R.; Tuminaro, R., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, J. Comput. Phys., 227, 3, 1790-1808 (2008) · Zbl 1290.76023
[42] Knoll, D. A.; Mousseau, V. A., On Newton-Krylov multigrid methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 163, 1, 262-267 (2000) · Zbl 0994.76055
[43] Pernice, M.; Tocci, M. D., A multigrid-preconditioned Newton-Krylov method for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 23, 2, 398-418 (2001) · Zbl 0995.76061
[44] Balay, S.; Eijkhout, V.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., Efficient management of parallelism in object oriented numerical software libraries, (Arge, E.; Bruaset, A. M.; Langtangen, H. P., Modern Software Tools in Scientific Computing (1997), Birkhäuser Press), 163-202 · Zbl 0882.65154
[45] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2009. <http://www.mcs.anl.gov/petsc>; S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2009. <http://www.mcs.anl.gov/petsc>
[46] S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc users manual, Tech. Rep. ANL-95/11 - Revision 3.0.0, Argonne National Laboratory, 2008.; S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc users manual, Tech. Rep. ANL-95/11 - Revision 3.0.0, Argonne National Laboratory, 2008.
[47] Gresho, P. M.; Sani, R. L., Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow (1998), John Wiley & Sons · Zbl 0941.76002
[48] Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid, 8, 12, 2182-2189 (1965) · Zbl 1180.76043
[49] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 2, 461-469 (1993) · Zbl 0780.65022
[50] Simoncini, V.; Szyld, D. B., Flexible inner-outer Krylov subspace methods, SIAM J. Numer. Anal., 40, 6, 2219-2239 (2003) · Zbl 1047.65021
[51] Simoncini, V.; Szyld, D. B., Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14, 1, 1-59 (2007) · Zbl 1199.65112
[52] Murphy, M. F.; Golub, G.; Wathen, A. J., A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21, 6, 1969-1972 (2000) · Zbl 0959.65063
[53] Schaffer, S., A semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients, SIAM J. Sci. Comput., 20, 1, 228-242 (1998) · Zbl 0913.65111
[54] Brown, P. N.; Falgout, R. D.; Jones, J. E., Semicoarsening multigrid on distributed memory machines, SIAM J. Sci. Comput., 21, 5, 1823-1834 (2000), also available as LLNL technical report UCRL-JC-130720 · Zbl 0958.65134
[55] Falgout, R. D.; Jones, J. E., Multigrid on massively parallel architectures, (Dick, E.; Riemslagh, K.; Vierendeels, J., Multigred Methods VI. Multigred Methods VI, Lecture Notes in Computational Science and Engineering, vol. 14 (2000), Springer-Verlag), 101-107, also available as LLNL Technical Report UCRL-JC-133948. · Zbl 0972.65110
[56] Ashby, S. F.; Falgout, R. D., A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations, Nucl. Sci. Eng., 124, 1, 145-159 (1996), also available as LLNL Technical Report UCRL-JC-122359
[57] hypre<http://www.llnl.gov/CASC/hypre>; hypre<http://www.llnl.gov/CASC/hypre> · Zbl 1056.65046
[58] R.D. Falgout, U.M. Yang, hypre; R.D. Falgout, U.M. Yang, hypre · Zbl 1056.65046
[59] Peng, Y.-F.; Shiau, Y. H.; Hwang, R. R., Transition in a 2-D lid-driven cavity flow, Comput. Fluid, 32, 3, 337-352 (2003) · Zbl 1009.76513
[60] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2002) · Zbl 1123.74309
[61] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 2, 614-648 (2001) · Zbl 1157.76369
[62] Toth, G.; Roe, P. L., Divergence- and curl-preserving prolongation and restriction formulas, J. Comput. Phys., 180, 2, 736-750 (2002) · Zbl 1143.65322
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