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Approximate factorization as a high order splitting for the implicit incompressible flow equations. (English) Zbl 0760.76059
Summary: We apply the method of approximate factorization to derive a second-order accurate splitting of the incompressible flow equations (Stokes or Navier-Stokes equations). This is novel because the method of approximate factorization was believed inapplicable to this type of equation system. We demonstrate the resulting splitting on a second-order Crank-Nicolson discretization and point out its intimate relationship to some existing second-order accurate splitting (projection) methods. Further, we use the approximate factorization method to derive entirely new splittings. We first develop a new generalized second-order accurate splitting which may be specialized to a variety of applications. We indicate its applications to finite-elements, “checkerboard-free” cell-centered discretizations, and ocean modeling. We then generalize the original splitting to an arbitrary high-order scheme.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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[1] Harlow, F.H.; Welch, J.E., Phys. fluids, 8, 2182, (1965)
[2] Chorin, A.J., Math. comput., 22, 745, (1968)
[3] Yanenko, N.N., The method of fractional steps, (1971), Springer-Verlag New York · Zbl 0209.47103
[4] Gresho, P.M., Int. J. num. methods fluids, 11, 587, (1990)
[5] Orszag, S.A.; Israeli, M.; Deville, M.O., J. sci. comput., 1, 75, (1986)
[6] Gresho, P.M.; Sani, R., Int. J. num. methods fluids, 7, 1111, (1987)
[7] Fortin, M.; Peyret, R.; Temam, R., J. mec., 10, 357, (1971)
[8] Kim, J.; Moin, P., J. comput. phys., 59, 308, (1985)
[9] Zang, T.A.; Hussaini, M.Y., Appl. math. comput., 19, 359, (1986)
[10] Deville, M.; Kleiser, L.; Montigny-Rannou, F., Int. J. num. methods fluids, 4, 1149, (1984)
[11] Van Kan, J., SIAM J. sci. stat. comput., 7, 870, (1986)
[12] Bell, J.B.; Colella, P.; Glaz, H.M., J. comput. phys., 85, 257, (1989)
[13] Warming, R.F.; Beam, R.M., (), 85
[14] Marchuk, G.I., Methods of numerical mathematics, (1982), Springer-Verlag New York · Zbl 0485.65003
[15] Briley, W.R.; McDonald, H., J. comput. phys., 34, 54, (1980)
[16] Briley, W.R.; McDonald, H., J. comput. phys., 24, 372, (1977)
[17] Gresho, P.M.; Chan, S.T., Int. J. num. methods fluids, 11, 621, (1990)
[18] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., J. comput. phys., 97, 414, (1991)
[19] A.S. Dvinsky and J.K. Dukowicz, Computers and Fluids, submitted.
[20] Shaw, C.T., Int. J. num. methods fluids, 12, 81, (1991)
[21] Rice, J.G.; Schnipke, R.J., Comput. methods appl. mech. eng., 58, 135, (1986)
[22] Semtner, A.J., Advanced physical oceanographic numerical modelling, (), 187
[23] Hildebrand, F.B., Introduction to numerical analysis, (1982), McGraw-Hill New York · Zbl 0070.12401
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