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Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations. (English) Zbl 0846.76075
This paper examines the long-term behavior, dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible Navier-Stokes equations, including both direct schemes and fractional step/projection methods. Numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
76B47 Vortex flows for incompressible inviscid fluids
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Temam, R., Navier-Stokes equations and nonlinear functional analysis, () · Zbl 0522.35002
[2] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, () · Zbl 0871.35001
[3] Constantin, P.; Foias, C., Navier-Stokes equations, (1988), Univ. of Chicago Press Chicago, IL · Zbl 0687.35071
[4] Shen, J., Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. anal., 38, 201-229, (1990) · Zbl 0684.65095
[5] Foias, C.; Jolly, M.S.; Kevrekidis, I.G.; Titi, E.S., Dissipativity of numerical schemes, Nonlinearity, 4, 591-613, (1991) · Zbl 0734.65080
[6] Elliot, C.M.; Stuart, A.M., The global dynamics of discrete semilinear equations, () · Zbl 0792.65066
[7] Simo, J.C.; Tarnow, N., The discrete energy-momentum method. conserving algorithms for nonlinear elastodynamics, Z. angew. math., (1992), in press. · Zbl 0758.73001
[8] Ge, Z.; Marsden, J.E., Lie-Poisson Hamilton-jacob; theory and Lie-Poisson integrators, Phys. lett. A, 133, 134-139, (1988) · Zbl 1369.70038
[9] D. Lewis and J.C. Simo, Conserving algorithm for the dynamics of Hamiltonian systems on Lie groups, J. Nonlinear Sci., submitted. · Zbl 0799.58069
[10] Scovel, C., Symplectic numerical integration of Hamiltonian systems, (), 463-496 · Zbl 0748.58010
[11] Hughes, T.J.R.; Liu, W.K.; Brooks, A., Review of finite element analysis of incompressible viscous flow by the penalty function formulation, J. comput. phys., 30, 1-60, (1979)
[12] Chorin, A.E., A numerical method for solving incompressible viscous flow problems, J. comput. phys., 2, 12-26, (1967) · Zbl 0149.44802
[13] Chorin, A.E., Numerical solution of the Navier-Stokes equations, Math. comp., 22, 745-762, (1968) · Zbl 0198.50103
[14] Chorin, A.E., On the convergence of discrete approximations to the Navier-Stokes equations, Math. comp., 23, 341-353, (1969) · Zbl 0184.20103
[15] Hestenes, M.R., Multiplier and gradient methods, J. optim. theory appl., 4, 303-320, (1969) · Zbl 0174.20705
[16] Powell, M.J.D., A method for nonlinear constraints in minimization problems, () · Zbl 0194.47701
[17] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer Berlin · Zbl 0575.65123
[18] Glowinski, R.; Le Tallec, P., Augmented Lagrangian and operator splitting methods in nonlinear mechanics, () · Zbl 0698.73001
[19] Simo, J.C.; Taylor, R.L., Quasi-incompressible finite elasticity in principal stretches; continuum basis and numerical algorithms, Comput. methods appl. mech. engrg., 85, 273-310, (1991) · Zbl 0764.73104
[20] Yanenko, N.N., Fractional step methods, (1971), Springer Berlin, English translation · Zbl 0209.47103
[21] Temam, R., Une méthode d’aproximations de la solution des équations de Navier-Stokes, Bull. soc. math. France, 98, 115-152, (1968) · Zbl 0181.18903
[22] Temam, R., Navier-Stokes equations, (1979), North-Holland Amsterdam · Zbl 0454.35073
[23] Kim, J.; Moin, P., Application of a fractional-step methods to the incompressible Navier-Stokes equations, J. comput. phys., 59, 308-323, (1985) · Zbl 0582.76038
[24] Bell, J.; Collela, P.; Glaz, H., A second-order projection method for the incompressible Navier-Stokes equations, J. comput. phys., (1990), to appear.
[25] Shen, J., On error estimates of projection methods for Navier-Stokes equations: first order schemes, SIAM J. numer. anal., (1991), to appear.
[26] Gresho, P.M., Some current CFD issues relevant to the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 87, 201-252, (1991) · Zbl 0760.76018
[27] Kreiss, H.; Lorenz, J., Initial-boundary value problems and the Navier-Stokes equations, () · Zbl 1097.35113
[28] Chorin, A.E.; Marsden, J.E., A mathematical introduction to fluid mechanics, (1990), Springer Berlin · Zbl 0712.76008
[29] Mahalov, Titi and Leibovich (1990).
[30] Arnold, V.I., Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications á l’hydrodynamique des fluides perfaits, Ann. inst. Fourier, XVI, 319-361, (1966) · Zbl 0148.45301
[31] Arnold, V.I., Mathematical methods of classical mechanics, (1989), Springer Berlin
[32] Ebin, D.; Marsden, J.E., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. math., 92, 102-163, (1970) · Zbl 0211.57401
[33] Dahlquist, G., Error analysis of a class of methods for stiff nonlinear initial value problems, (), 60-74
[34] Butcher, J.C., A stability property of implicit Runge-Kutta methods, Bit, 15, 358-361, (1975) · Zbl 0333.65031
[35] Simo, J.C., Nonlinear stability analysis of the time-discrete variational problem of evolution in nonlinear heat conduction, plasticity and viscoplasticity, Comput. methods appl. mech. engrg., 88, 111-131, (1991) · Zbl 0751.73066
[36] A. Stuart and A.R. Humphries, Model problems in numerical stability theory of initial boundary value problems, SIAM Rev. (preprint). · Zbl 0807.65091
[37] Strang, G.; Fix, D., An analysis of the finite element method, (1973), Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[38] Pironneau, O., Finite elements for fluids, (1989), Wiley New York · Zbl 0665.73059
[39] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Berlin · Zbl 0788.73002
[40] Crouzeix, M.; Raviart, P.A., Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO anal. numer., 7, 33-76, (1973) · Zbl 0302.65087
[41] Fortin, M.; Fortin, A., A generalization of Uzawa’s algorithm for the Navier-Stokes equations, Comm. appl. numer. methods, 1, 205-208, (1985) · Zbl 0592.76040
[42] Fortin et al. (1971).
[43] 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, 387-411, (1982) · Zbl 0511.76031
[44] Brezzi, F.; Bristeau, M.-O.; Franca, L.P.; Mallet, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. methods appl. mech. engrg., 96, 117-229, (1992) · Zbl 0756.76044
[45] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[46] Johnson, C.; Nävert, U.; Pitkaranta, J., Finite element methods for linear hyperbolic systems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[47] Gresho, P.M.; Lee, R.L.; Chan, S.T.; Sani, R.L., Solution of the time-dependent incompressible Navier-Stokes equations using the Galerkin finite element method, (), 203-222
[48] Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations, ()
[49] Roshko, A., On the development of turbulent wakes from vortex streets, NACA report 1194, (1954)
[50] ()
[51] Marsden, J.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice Hall Englewood Cliffs, NJ
[52] Sauer, T.; Yorke, J.A.; Casdagli, M., Embeddology, J. statist. phys., 65, 579-616, (1991)
[53] Lee, H.; Moin, P., An improvement of fractional-step methods for the incompressible Navier-Stokes equations, J. comput. phys., 92, 369-379, (1991) · Zbl 0709.76030
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