High-order compact scheme for the steady stream-function vorticity equations. (English) Zbl 0836.76065

Summary: A higher-order compact scheme that is \(O(h^4)\) on the nine-point two- dimensional stencil is formulated for the steady stream-function vorticity form of the Navier-Stokes equations. The resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software. We also show that special treatment of the wall boundary conditions is required. The method is tested on representative model problems and compares very favourably with other schemes in the literature.


76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Leonard, Comput. Methods Appl. Mech. Eng. 19 pp 59– (1979)
[2] Leonard, Int. j. numer. methods eng. 30 pp 729– (1990)
[3] Noye, Int. j. numer. methods eng. 26 pp 1615– (1988)
[4] Noye, Commun. appl. numer. methods 6 pp 279– (1990)
[5] MacKinnon, J. Comput. Phys. 75 pp 151– (1988)
[6] MacKinnon, Int. j. numer. methods eng. 28 pp 489– (1989)
[7] MacKinnon, SIAM J. Scientific Statist. Comput. 11 pp 343– (1990)
[8] MacKinnon, Commun., appl. numer. methods 6 pp 47– (1990)
[9] MacKinnon, Int. j. numer. methods fluids 13 pp 739– (1991)
[10] . ’Superconvergent finite difference methods with applications to viscous flow’, Master’s thesis, University of Texas at Austin, 1991.
[11] Johnson, Commun. appl. numer. methods 8 pp 99– (1992)
[12] Johnson, Commun. appl. numer. methods 8 pp 841– (1992)
[13] Dukowitz, J. Comput. Phys. 32 pp 71– (1979)
[14] . ’Mixed defect correction iteration for the accurate solution of the convection diffusion equation’, in and (eds.), Multigrid Methods, Proceedings of Conference held in Köln-Porz, 23–27 November 1981, Springer, Berlin, 1982, pp. 485–501.
[15] . et al., ’Single-cell high order difference method for steady state advection-diffusion equation’, in Proc. Symp. Int. Assoc. for Hydraulic Research, 1982, pp. 101–108.
[16] Gupta, Int. j. numer. methods fluids 4 pp 641– (1984)
[17] Dennis, J. Comput. Phys. 85 pp 390– (1989)
[18] Abarbanel, J. Scientific Comput. 3 pp 275– (1988)
[19] . and , ”Formulation and experiments with high-order compact schemes for nonuniform grids”, submitted to IMA Journal of Numerical Analysis, 1994.
[20] . Singularities and Constructive Methods for Their Treatment, Proceedings of the Conference held in Oberwolfach, West Germany, Springer, New York, 1985.
[21] . and , ”High Re solutions for incompressible flow using Navier-Stokes equations and a multi-grid method”, J. Comput. Phys. 387–411 (1982). · Zbl 0511.76031
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