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An overview of projection methods for incompressible flows. (English) Zbl 1122.76072
Summary: We address a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows. These methods are often referred to in the literature as projection methods, and can be classified into three classes, namely the pressure-correction methods, velocity-correction methods, and consistent splitting methods. For each class of schemes, theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed. The essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners.

MSC:
76M99 Basic methods in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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[1] Achdou, Y.; Guermond, J.-L., Convergence analysis of a finite element projection/lagrange – galerkin method for the incompressible navier – stokes equations, SIAM J. numer. anal., 37, 3, 799-826, (2000) · Zbl 0966.76041
[2] Auteri, F.; Guermond, J.-L.; Parolini, N., Role of the LBB condition in weak spectral projection methods, J. comput. phys., 174, 405-420, (2001) · Zbl 1007.76056
[3] Batoul, A.; Khallouf, H.; Labrosse, G., Une méthode de résolution directe (pseudo-spectrale) du problème de Stokes 2D/3D instationnaire. application à la cavit’e entrainée carrée, CR acad. sci. Paris, Série I, 319, 1455-1461, (1994) · Zbl 0816.76062
[4] Bernardi, C.; Maday, Y., Approximations spectrales de problèmes aux limites elliptiques, (1992), Springer-Verlag Paris · Zbl 0773.47032
[5] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York · Zbl 0788.73002
[6] 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
[7] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York · Zbl 0658.76001
[8] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103
[9] Dauge, M., Stationary Stokes and navier – stokes systems on two- or three-dimensional domains with corners. I. linearized equations, SIAM J. math. anal., 20, 1, 74-97, (1989) · Zbl 0681.35071
[10] E, W.; Liu, J.G., Projection method I: convergence and numerical boundary layers, SIAM J. numer. anal., 32, 1017-1057, (1995) · Zbl 0842.76052
[11] E, W.; Liu, J.G., Gauge method for viscous incompressible flows, Commun. math. sci., 1, 2, 317-332, (2003) · Zbl 1160.76329
[12] Girault, V.; Raviart, P.-A., Finite element methods for navier – stokes equations, (1986), Springer-Verlag Berlin, Theory and algorithms · Zbl 0413.65081
[13] Glowinski, R., Finite element methods for incompressible viscous flow, Handbook of numerical analysis, vol. IX, (2003), Elsevier Science B.V. Amsterdam · Zbl 1040.76001
[14] Goda, K., A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows, J. comput. phys., 30, 76-95, (1979) · Zbl 0405.76017
[15] Gresho, P.M.; Chan, S.T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via finite element method that also introduces a nearly consistent mass matrix. part I and part II, Int. J. numer. methods fluids, 11, (1990), 587-620, 621-659 · Zbl 0712.76035
[16] Grisvard, P., Boundary value problems in non-smooth domains, (1985), Pitman · Zbl 0695.35060
[17] Guermond, J.-L., Sur l’approximation des équations de navier – stokes par une méthode de projection, CR acad. sci. Paris, Série I, 319, 887-892, (1994) · Zbl 0813.76066
[18] Guermond, J.-L., Some practical implementations of projection methods for navier – stokes equations, Modél. math. anal. num., 30, 637-667, (1996) · Zbl 0861.76065
[19] Guermond, J.-L., Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de navier – stokes par une technique de projection incrémentale, M2AN math. model. numer. anal., CR acad. sci. Paris, Série I, 325, 1, 1329-1332, (1997), Also in · Zbl 0899.76271
[20] Guermond, J.-L.; Minev, P., Analysis of a projection/characteristic scheme for incompressible flow, Comm. numer. methods engrg., 19, 535-550, (2003) · Zbl 1112.76389
[21] Guermond, J.-L.; Minev, P.; Shen, J., Error analysis of pressure-correction schemes for the navier – stokes equations with open boundary conditions, SIAM J. numer. anal., 43, 1, 239-258, (2005) · Zbl 1083.76044
[22] Guermond, J.-L.; Quartapelle, L., On stability and convergence of projection methods based on pressure Poisson equation, Int. J. numer. methods fluids, 26, 9, 1039-1053, (1998) · Zbl 0912.76054
[23] Guermond, J.-L.; Quartapelle, L., On the approximation of the unsteady navier – stokes equations by finite element projection methods, Numer. math., 80, 5, 207-238, (1998) · Zbl 0914.76051
[24] Guermond, J.L.; Shen, J., On the error estimates for the rotational pressure-correction projection methods, Math. comput., 73, 248, 1719-1737, (2004), (electronic) · Zbl 1093.76050
[25] Guermond, J.L.; Shen, J., Quelques résultats nouveaux sur LES méthodes de projection, CR acad. sci. Paris, Série I, 333, 1111-1116, (2001) · Zbl 1078.76587
[26] Guermond, J.L.; Shen, J., A new class of truly consistent splitting schemes for incompressible flows, J. comput. phys., 192, 262-276, (2003) · Zbl 1032.76529
[27] Guermond, J.L.; Shen, J., Velocity-correction projection methods for incompressible flows, SIAM J. numer. anal., 41, 1, 112-134, (2003) · Zbl 1130.76395
[28] Heywood, J.G.; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible navier – stokes equations, Int. J. numer. methods fluids, 22, 5, 325-352, (1996) · Zbl 0863.76016
[29] Hugues, S.; Randriamampianina, A., An improved projection scheme applied to pseudospectral methods for the incompressible navier – stokes equations, Int. J. numer. methods fluids, 28, 3, 501-521, (1998) · Zbl 0932.76065
[30] Johnston, H.; Liu, J.-G., Accurate, stable and efficient navier – stokes solvers based on explicit treatment of the pressure term, J. comput. phys., 199, 1, 221-259, (2004) · Zbl 1127.76343
[31] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J. comput. phys., 97, 414-443, (1991) · Zbl 0738.76050
[32] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for CFD, (1999), Oxford University Press · Zbl 0954.76001
[33] Kellogg, R.B.; Osborn, J.E., A regularity result for the Stokes problem in a convex polygon, J. funct. anal., 21, 4, 397-431, (1976) · Zbl 0317.35037
[34] 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
[35] Ladyzhenskaya, O.A., The mathematical theory of viscous incompressible flow. second English edition, revised and enlarged. translated from the Russian by richard A. silverman and John Chu, Mathematics and its applications, vol. 2, (1969), Gordon and Breach Science Publishers New York
[36] Lee, M.J.; Oh, B.D.; Kim, Y.B., Canonical fractional-step methods and consistent boundary conditions for the incompressible navier – stokes equations, J. comput. phys., 168, 73-100, (2001) · Zbl 1074.76585
[37] Leriche, E.; Labrosse, G., High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties, SIAM J. sci. comput., 22, 4, 1386-1410, (2000), (electronic) · Zbl 0972.35087
[38] Minev, P., A stabilized incremental projection scheme for the incompressible navier – stokes equations, Int. J. numer. methods fluids, 36, 441-464, (2001) · Zbl 1035.76028
[39] Orszag, S.A.; Israeli, M.; Deville, M., Boundary conditions for incompressible flows, J. sci. comput., 1, 75-111, (1986) · Zbl 0648.76023
[40] Perot, J.B., An analysis of the fractional step method, J. comput. phys., 108, 1, 51-58, (1993) · Zbl 0778.76064
[41] Prohl, A., Projection and quasi-compressibility methods for solving the incompressible navier – stokes equations, Advances in numerical mathematics, (1997), B.G. Teubner Stuttgart · Zbl 0874.76002
[42] Quarteroni, A.; Saleri, F.; Veneziani, A., Analysis of the Yosida method for the incompressible navier – stokes equations, J. math. pures appl. (9), 78, 5, 473-503, (1999) · Zbl 0930.35127
[43] Quarteroni, A.; Saleri, F.; Veneziani, A., Factorization methods for the numerical approximation of navier – stokes equations, Comput. methods appl. mech. engrg., 188, 1-3, 505-526, (2000) · Zbl 0976.76044
[44] R. Rannacher, On Chorin’s projection method for the incompressible Navier-Stokes equations, in: Lecture Notes in Mathematics, vol. 1530, 1991. · Zbl 0769.76053
[45] Sacchi-Landriani, G.; Vandeven, H., Polynomial approximation of divergence-free functions, Math. comput., 52, 103-130, (1989) · Zbl 0694.41009
[46] Shen, J., On error estimates of the projection methods for the navier – stokes equations: first-order schemes, SIAM J. numer. anal., 29, 57-77, (1992) · Zbl 0741.76051
[47] Shen, J., Efficient spectral-Galerkin method. I. direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. sci. comput., 15, 6, 1489-1505, (1994) · Zbl 0811.65097
[48] Shen, J., On error estimates of projection methods for the navier – stokes equations: second-order schemes, Math. comput., 65, 215, 1039-1065, (1996) · Zbl 0855.76049
[49] Shen, J., A new pseudo-compressibility method for the navier – stokes equations, Appl. numer. math., 21, 71-90, (1996)
[50] Strikwerda, J.C.; Lee, Y.S., The accuracy of the fractional step method, SIAM J. numer. anal., 37, 1, 37-47, (1999) · Zbl 0953.65061
[51] Temam, R., Sur l’approximation de la solution des équations de navier – stokes par la méthode des pas fractionnaires ii, Arch. ration. mech. anal., 33, 377-385, (1969) · Zbl 0207.16904
[52] Temam, R., Navier – stokes equations: theory and numerical analysis, (1984), North-Holland Amsterdam · Zbl 0568.35002
[53] Timmermans, L.J.P.; Minev, P.D.; Van De Vosse, F.N., An approximate projection scheme for incompressible flow using spectral elements, Int. J. numer. methods fluids, 22, 673-688, (1996) · Zbl 0865.76070
[54] van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. sci. stat. comput., 7, 3, 870-891, (1986) · Zbl 0594.76023
[55] Wang, C.; Liu, J.-G., Convergence of gauge method for incompressible flow, Math. comput., 69, 232, 1385-1407, (2000) · Zbl 0968.76065
[56] Wheeler, M.F., A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. numer. anal., 10, 723-759, (1973) · Zbl 0232.35060
[57] Yanenko, N.N., The method of fractional steps. the solution of problems of mathematical physics in several variables, (1971), Springer-Verlag New York · Zbl 0209.47103
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