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On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. I: Theory. (English) Zbl 0712.76035

Summary: Ever since the time of A. J. Chorin’s classic paper [Math. Comput. 22, 745- 762 (1968; Zbl 0198.501)] on projection methods, there have been lingering and poorly understood issues related to the best - or even proper or appropriate - boundary conditions (BCs) that should be (or could be) applied to the ‘intermediate’ velocity when the viscous terms in the incompressible Navier-Stokes equations are treated with an implicit time integration method and a Poisson equation is solved as part of a ‘time step’. These issues also pervade all related methods that uncouple the equations by ‘splitting’ the pressure computation from that of the velocity - at least in the presence of solid boundaries and (again) when implicit treatment of the viscous terms is employed. This paper is intended to clarify these issues by showing which intermediate BCs are ‘best’ and why some that are not work well anyway. In particular we show that all intermediate BCs must cause problems related to the regularity of the solution near boundaries, but that a near-miraculous recovery occurs such that accurate results are nevertheless achieved beyond the spurious boundary layer introduced by such methods. The mechanism for this ‘miracle’ is related to the existence of a higher- order equation that is actually satisfied by the pressure. All that is required then for projection (splitting, fractional step, etc.) methods to work well is that the spurious boundary layer be thin - as has been largely observed in practice.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Gresho, Int. j. numer. methods fluids 7 pp 1111– (1987)
[2] Chorin, Math. Comput. 22 pp 745– (1968)
[3] Chorin, Math. Comput. 23 pp 341– (1969)
[4] Orszag, J. Sci. Comput. 1 pp 75– (1986)
[5] and , ’Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations’, LLNL Report UCRL-99503, 1988.
[6] and , ’Solving the incompressible Navier-Stokes equations using consistent mass and a pressure Poisson equation’, Proc. ASME Symp. on Recent Advances in Computational Fluid Dynamics, AMD-Vol. 95, pp. 51-74, Chicago, IL, 28 November-2 December 1988.
[7] and , ’Accurate explicit finite element schemes for convective-conductive heat transfer problems’, in Finite Element Methods for Convection-Dominated Flows, AMD-Vol. 34, ASME, N.Y., 1979. pp. 149-166,
[8] , and , ’Solution of the unsteady Navier-Stokes equations by a finite element projection method’, in and (eds), Computational Techniques in Transient and Turbulent Flow, Vol. 2, pp. 97-132, Pineridge Press, Swansea, 1981.
[9] Donea, Comput. Methods Appl. Mech. Eng. 30 pp 53– (1982)
[10] and , ’Numerical solution of viscous incompressible flows’, Annual Review of Fluid Mechanics, Annual Reviews, Palo Alto, CA, 1974, pp. 281-319.
[11] Heywood, Indiana Univ. Math. J. 29 pp 639– (1980)
[12] Heywood, SIAM J. Numer Anal. 19 pp 73– (1982)
[13] Strang, SIAM Rev. 30 pp 283– (1988)
[14] and , A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1979. · Zbl 0417.76002
[15] Kim, J. Comput. Phys. 59 pp 308– (1985)
[16] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984)
[17] Fortin, J. Méc. 10 pp 357–
[18] Zang, Appl. Math. Comput. 19 pp 359– (1986)
[19] and , Incompressible Flows and the Finite Element Method, Wiley, Chichester 1991.
[20] Van Kan, SIAM J. Sci. Stat. Comput. 7 pp 870– (1986)
[21] and , Personal communication, 1987.
[22] and , ’A second-order projection method for the incompressible Navier-Stokes equations’, LLNL Report UCRL-98225, 1988; submitted to J. Comput. Phys.
[23] Navier-Stokes Equations, North-Holland (Elsevier), Amsterdam, 1984.
[24] and , Numerical Analysis of Spectral Methods, SIAM, Philadelphia, PA, 1977.
[25] Orszag, J. Fluid Mech. 96 pp 159– (1980)
[26] Marcus, J. Fluid Mech. 146 pp 45– (1984)
[27] Amsden, J. Comput. Phys. 6 pp 322– (1970)
[28] and , Computational Methods for Fluid Flow, Springer, New York, 1983. · Zbl 0514.76001
[29] Easton, J. Comput. Phys. 9 pp 375– (1972)
[30] Personal communication, 1987.
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