Abstract
A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC 3 continuity is speculated to be its cause.
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References
Boyd, J. P. (1989).Chebyshev and Fourier Spectral Methods, Lecture Notes in Engineering, Vol. 49. Springer-Verlag, New York.
Canuto, C., Hussaini, M. H., Quarteroni, A., and Zang, T. A. (1988).Spectral Methods in Fluid Dynamics. Springer-Verlag, New York.
Chan, T. F. (1987). Analysis of preconditioners for domain decomposition.SIAM J. Num. Anal. 24, 382–390.
Ghia, U., Ghia, K. N., and Shin, C. T. (1982). High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,J. Comput. Phys. 48, 387–411.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia.
Johnson, G. M. (1987). Parallel processing in fluid dynamics, In O. Baysal (ed.),Applications of Parallel Processing in Fluid Mechanics, ASME, New York.
Karageorghis, A., Phillips, T. N., and Davies, A. R. (1988). Spectral collocation methods for the primary two-point boundary value problem in modeling viscoelastic flows,Int. J. Num. Methods Eng. 26, 805–813.
Keyes, E. D., and Gropp, W. D. (1987). A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation.SIAM J. Sci. Stat. Comput. 8, s166-s202.
Lai, C.-H. (1992). A nonoverlapped domain decomposition for a class of convection-diffusion problems,Appl. Math. Modeling 16, 101–106.
Mousseau, V. A., Nguyen, H. D., and Paik, S. (1993), unpublished results.
Nguyen, H. D., and Paik, S. (1993a). A non-iterative solution approach for parallel pseudospectral domain decomposition.J. Sci. Comput. 8, 357–372.
Nguyen, H. D., and Paik, S. (1993b). Solution domain decomposition method for parallel computational heat transfer, submitted toJ. Comput. Phys.
Orszag, S. A. (1971). Accurate solution of the Orr-Sommerfeld stability equation,J. Fluid Mech. 50, 689–703.
Phillips, T. N. (1984). Natural convection in an enclosed cavity,J. Comput. Phys. 54, 365–381.
Yousif, W. S., and Evans, D. J. (1993). Explicit block iterative method for the solution of the biharmonic equation,Num. Meth. Part. Diff. Eqs. 9, 1–12.
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Nguyen, H.D., Paik, S. Solution domain decomposition method for viscous incompressible flow. J Sci Comput 9, 351–368 (1994). https://doi.org/10.1007/BF01575038
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DOI: https://doi.org/10.1007/BF01575038