zbMATH — the first resource for mathematics

An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow. (English) Zbl 0724.76070
Summary: We present a simple, general methodology for the generation of high-order operator decomposition (“splitting”) techniques for the solution of time-dependent problems arising in ordinary and partial differential equations. The new approach exploits operator integration factors to reduce multiple-operator equations to an associated series of single- operator initial-value subproblems. Two illustrations of the procedure are presented: the first, a second-order method in time applied to velocity-pressure decoupling in the incompressible Stokes problem; the second, a third-order method in time applied to convection-Stokes decoupling in the incompressible Navier-Stokes equations. Critical open questions are briefly described.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Adams, R. A. (1975).Sobolev Spaces, Academic Press, New York. · Zbl 0314.46030
[2] Ames, W. F. (1977).Numerical Methods for Partial Differential Equations, Academic Press, New York. · Zbl 0577.65077
[3] Arrow, K., Hurwicz, L., and Uzawa, H. (1958).Studies in Nonlinear Programming, Stanford University Press, Stanford. · Zbl 0091.16002
[4] Benqué, J. P., Ibler, B., Keramsi, A., and Labadie, G. (1982). A new finite element method for Navier-Stokes equations coupled with a temperature equation, inProc. 4th Int. Symp. on Finite Elements in Flow Problems, Kawai, T. (ed). North-Holland, Amsterdam, pp. 295-301.
[5] Bristeau, M. O., Glowinski, R., and Periaux, J. (to appear). Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incompressible viscous flows.Comput. Phys. Rep. · Zbl 0675.76030
[6] Brooks, A. N., and Hughes, T. J. R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible · Zbl 0497.76041
[7] Navier-Stokes equations,Comput. Methods Appl. Meck Eng. 32, 199-259. · Zbl 0497.76041
[8] Bussing, T. R. A., and Murman, E. M. (1988). Finite volume method for the calculation of compressible chemically reacting flows, inAIAA J. 26, 1070-1078. · Zbl 0661.76117
[9] Cahouet, J., and Chabard, J. P. (1988). Some fast three-dimensional finite element solvers for the generalized Stokes problem,Int. J. Numer. Methods Fluids 8, 869-895. · Zbl 0665.76038
[10] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988).Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin. · Zbl 0658.76001
[11] Chao, Y., and Attard, A. (1985). A resolution of the stiffness problem of reactor kinetics,Nucl. Sci. Eng. 90, 40-46.
[12] Chorin, A. J. (1970). Numerical solution of incompressible flow problems, inStudies in Numerical Analysis 2, Ortega, J. M., and Rheinboldt, W. C. (eds.), SIAM, Philadelphia. · Zbl 0225.76017
[13] Deville, M. O., Kleiser, L., and Montigny-Rannou, F. (1984). Pressure and time treatment for Chebyshev spectral solution of a Stokes problem,Int. J. Methods Fluids 4, 1149. · Zbl 0554.76033
[14] Douglas, J., Jr. (1955).SIAM 3, 42.
[15] Ewing, R. E., and Russell, T. F. (1981). Multistep Galerkin methods along characteristics for convection-diffusion problems, inAdvances in Computer Methods for Partial Differential Equations-IV, Vichnevetsky, R., and Stepleman, R. S. (eds.), IMACS, Rutgers University, New Brunswick, New Jersey, pp. 28-36.
[16] Gear, C. W. (1971). Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey. · Zbl 1145.65316
[17] Gear, C. W. (1980). Automatic multirate methods for ordinary differential equations,International Federation for Information Processing, North-Holland, Amsterdam.
[18] Girault, V., and Raviart, P. A. (1986).Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin. · Zbl 0585.65077
[19] Glowinski, R. (1984).Numerical Methods for Nonlinear Variational Problems, SpringerVerlag, Berlin. · Zbl 0536.65054
[20] Golub, G. H., and Van Loan, C. F. (1983).Matrix Computations, John Hopkins University Press, Baltimore, Maryland. · Zbl 0559.65011
[21] Gresho, P. M., Chan, S. T., Lee, R. L., and Upson, C. D. (1984). A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations, Part 1: Theory,Int. J. Numer. Methods Fluids 4, 557-598. · Zbl 0559.76030
[22] Friesner, R. A., Tuckerman, L. S., Dornblaser, B. C., and Russo, T. V. (to appear). A method for exponential propagation of large systems of stiff nonlinear differential equations, in Proceedings of the Second Nobeyama Workshop on Supercomputers and Fluid Dynamics (Japan, 1989),J. Sci. Comput.
[23] Ho, L. W. (1989). A Spectral Element Stress Formulation of the Navier-Stokes Equations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge.
[24] Ho, L. W., Maday, Y., Patera, A. T., and Rønquist, E. M. (1990). A high-order Lagrangiandecoupling method for the incompressible Navier-Stokes equations,Comput. Methods Appl. Mech. Eng. (to appear). · Zbl 0722.76038
[25] Ho, L. W., and Patera, A. T. (1990a). A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows,Comput. Methods Appl. Mech. Eng. (to appear). · Zbl 0722.76052
[26] Ho, L. W., and Patera, A. T. (1990b). Variational formulation of three-dimensional viscous free-surface flows: Natural imposition of surface tension boundary conditions,Int. J. Numer. Methods Fluids (to appear). · Zbl 0739.76057
[27] Hofer, E. (1976). A partially implicit method for large stiff systems of ODE’s with only a few equations introducing small time-constants,SIAM J. Numer. Anal. 13(5), 645. · Zbl 0399.65043
[28] Karniadakis, G. E. (to appear). Spectral element simulations of laminar and turbulent flows in complex geometries,Appl. Numer. Math. · Zbl 0678.76050
[29] Karniadakis, G. E., Israeli, M., and Orszag, S.A. (submitted). High-order splitting methods for the incompressible Navier-Stokes equations.J. Comput. Phys. · Zbl 0738.76050
[30] Karniadakis, G. E., Mikic, B. B., and Patera, A. T. (1988). Minimum-dissipation transport enhancement by flow destabilization: Reynolds’ analogy revisited,J. Fluid Mech. 192, 365-391.
[31] Kim, J., and Moin, P. (1985). Application of a fractional-step method to incompressible Navier-Stokes equations,J. Comput. Phys. 59, 308-323. · Zbl 0582.76038
[32] Kovasznay, L. I. G. (1948). Laminar flow behind a two-dimensional grid. InProc. Cambridge Phil. Soc. 44, 58-62. · Zbl 0030.22902
[33] Maday, Y., Meiron, D., Patera, A. T., and Rønquist, E. M. (submitted). Analysis of iterative method for the steady and unsteady Stokes problem: Application to spectral element discretizations. · Zbl 0769.76047
[34] Maday, Y., and Patera, A. T. (1989). Spectral element methods for the incompressible Navier-Stokes equations, inState of the Art Surveys in Computational Mechanics, Noor, A. K. (ed.). ASME, New York, pp. 71-143.
[35] Maday, Y., Patera, A. T., and Rønquist, E. M. (1987). A well-posed optimal spectral element approximation for the Stokes problem, ICASE Report No. 87-48.
[36] Marchuk, G. (1971). On the theory of the splitting method. InNumerical Solution of Partial Differential Equations II, Hubbard, B. (ed). Academic Press, New York, p. 469. · Zbl 0245.65045
[37] Orszag, S. A., Israeli, M., and Deville, M. O. (1986). Boundary conditions for incompressible flows,J. Sci. Comput. 1, 75. · Zbl 0648.76023
[38] Palusinski, O. A., and Wait, J. V. (1978). Simulation methods for combined linear and nonlinear systems,Simulation 30(3), 85-94. · Zbl 0379.65041
[39] Peaceman, D. W., and Rachford, H. H., Jr. (1955).SIAM 3, 28.
[40] Pironneau, O. (1982). On the transport-diffusion algorithm and its applications to the Navier-Stokes equations2,Numer. Math. 38, 309-332. · Zbl 0505.76100
[41] Rogallo, R. S. (1977). An ILLIAC Program for the Numerical Simulation of Homogeneous Incompressible Turbulence. NASA TM-73203.
[42] Rønquist, E. M. (1988). Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier-Stokes Equations, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge. · Zbl 0684.76030
[43] Saleh, R. A., and Newton, A. R. (1989). The exploitation of latency and multirate behavior using nonlinear relaxation for circuit simulation,IEEE Trans. Computer-Aided Design 8(12), 1286-1298. · Zbl 05449563
[44] Spalart, P. R. (1986). Numerical Simulation of Boundary Layers, Part 1: Weak Formulation and Numerical Method, NASA TM-88222. · Zbl 0617.76036
[45] Strang, G. (1968). On the construction and comparison of difference schemes,SIAM J. Numer. Anal. 5(3), 506-517. · Zbl 0184.38503
[46] Strang, G. (1990). Private communication.
[47] Temam, R. (1984).Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam. · Zbl 0568.35002
[48] Vandeven, H. (1990). Analysis of the eigenvalues of spectral differentiation operators, in Proceedings of the International Conference on Spectral and High-Order Methods for Partial Differential Equations,Comput. Methods Appl. Mech. Eng. (to appear). · Zbl 0727.65090
[49] Weidemann, J. A. C., and Trefethen, L. N. (1990). The eigenvalues of second-order spectral differentiation matrices.SIAM J. Numer. Anal, (to appear).
[50] White, J., and Vincentelli, A. S. (1987).Relaxation Techniques for the Simulation of VLSI Circuits, Kluwer Academic Press, Boston.
[51] Yanenko, N. N. (1971).The Method of Fractional Steps, Springer, New York. · Zbl 0209.47103
[52] Zang, T. A., and Hussaini, M. Y. (1986). On spectral multigrid methods for the timedependent Navier-Stokes equations,Appl. Math. Comput. 19, 359-372. · Zbl 0596.76032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.