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Parallel algorithms for the numerical simulation of three-dimensional natural convection. (English) Zbl 0719.76058

Summary: We deal with the numerical simulation of time-dependent three-dimensional natural convection on parallel computers working in single instruction multiple data (SIMD) mode. Applying finite differences on a staggered grid in combination with a pressure correction method to the underlying nonlinear system of partial differential equations, we reduce the numerical solution of the problem to the solution of a sequence of sparse linear systems. Using polynomial preconditioned conjugate gradient methods for the solution of these systems results in a highly parallel algorithm for the simulation of the considered flows on SIMD computers. Numerical experiments on an array processor illustrate the capabilities of the proposed algorithm.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76R10 Free convection

Software:

CGS
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References:

[1] Adams, L., M-step preconditioned conjugate gradient methods, SIAM J. Sci. Statist. Comput., 6, 452-463 (1985) · Zbl 0566.65018
[2] Axelsson, O.; Barker, V. A., Finite Element Solution of Boundary Value Problems (1984), Academic Press: Academic Press Orlando, FL · Zbl 0537.65072
[3] Bontoux, P.; Smutek, C.; Roux, B.; Lacroix, J. M., Three-dimensional buoyancy-driven flows in cylindrical cavities with differentially heated endwalls, Part 1: Horizontal cylinders, J. Fluid Mech., 169, 211-227 (1986)
[4] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 745-762 (1968) · Zbl 0198.50103
[5] Cuvelier, C.; Segal, A.; van Steenhoven, A. A., Finite Element Methods and Navier-Stokes Equations (1986), Reidel: Reidel Dordrecht · Zbl 0649.65059
[6] Harlow, F. J.; Welch, J. E., Numerical calculation of time dependent viscous incompressible flow of fluids with free surface, Phys. Fluid, 8, 2183-2189 (1960)
[7] Hestenes, M.; Stiefel, E., Methods of conjugate gradients for solving linear systems, Nat. Bur. Standards J. Res., 49, 409-436 (1952) · Zbl 0048.09901
[8] van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput., 891-8709 (1986) · Zbl 0594.76023
[9] Kirchartz, K. R.; Oertel, H., Three-dimensional thermal cellular convection in rectangular boxes, J. Fluid Mech., 192, 249-286 (1988)
[10] Mallinson, G. D.; de Vahl Davies, G., Three-dimensional natural convection in a box, J. Fluid Mech., 83, 1-31 (1977)
[11] Meijerink, J.; van der Vorst, H., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31, 148-162 (1977) · Zbl 0349.65020
[12] Meneguzzi, M.; Sulem, C.; Sulem, P. L.; Thual, O., Three-dimensional numerical simulation of convection in low-Prandtl-number fluids, J. Fluid Mech., 182, 169-191 (1987) · Zbl 0633.76049
[13] Müller, G.; Neumann, G.; Weber, W., Natural convection in vertical Bridgman configurations, J. Crystal Growth, 70, 78-93 (1984)
[14] Ortega, J.; Voigt, R., Solution of partial differential equations on vector and parallel computers, SIAM Rev., 27, 149-240 (1985) · Zbl 0644.65075
[15] Peyret, R.; Taylor, T. D., Computational Methods for Fluid Flow (1983), Springer: Springer Berlin · Zbl 0514.76001
[16] Rutishauser, H., Theory of gradient methods, (Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems (1959), Institute of Applied Mathematics: Institute of Applied Mathematics Zürich), 24-49
[17] Saad, Y., Practical use of polynomial preconditionings for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6, 865-881 (1985) · Zbl 0601.65019
[18] Smutek, C.; Roux, B.; Schiroky, G. H.; Hurford, A.; Rosenberger, F.; de Vahl Davies, G., Three-dimensional convection in horizontal cylinders—numerical solutions and comparison with experimental and analytical results, Numer. Heat Transfer, 8, 613-631 (1985)
[19] Sonneveld, P., CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10, 36-52 (1989) · Zbl 0666.65029
[20] Spiegel, E. A.; Veronis, G., On the Boussinesq approximation for a compressible fluid, Astrophys. J., 131, 442-447 (1960)
[21] Temam, R., Sur l’approximation des équations de Navier-Stokes par la méthode des pas fractionaires i, ii, Arch. Rat. Mech. Anal., 32, 377-385 (1969) · Zbl 0207.16904
[22] ten Thije Boonkkamp, J. H.M., The odd-even hopscotch pressure correction scheme for the incompressible Navier-Stokes equations, SIAM J. Sci. Statist. Comput., 9, 252-270 (1988) · Zbl 0651.76011
[23] Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Clarendon Press: Clarendon Press Oxford · Zbl 0258.65037
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