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High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

Published online by Cambridge University Press:  20 August 2015

Michel Fournié  and
Alain Rigal
Michel Fournié*
Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse et CNRS (UMR 5219), France
Alain Rigal*
Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse et CNRS (UMR 5219), France
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

Within the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-correction” method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.

Keywords

65M0676D05Incompressible Navier-Stokesfractional step methodhigh order compact schemeboundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 994 - 1019
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Chorin, A. J., The Numerical Solution of the Navier-Stokes Equations for an Incompressible Fluid, AEC Research and Developpement REPORT, NYO-1480-82, New York University, New York, 1967.Google Scholar
[2]Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdanm, 1979.Google Scholar
[3]Goda, K., A multistep technique with implicit difference scheme for calculating two or three dimentional cavity flows, J. Comput. Phys., 30 (1979), 76–95.CrossRefGoogle Scholar
[4]Kim, J., and Moin, P., Application of a fractional step method to imcompressible Navier-Stokes equations, J. Comput. Phys., 30 (1985), 308–323.Google Scholar
[5]Karniadakis, G. E., Israeli, M., and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414–443.CrossRefGoogle Scholar
[6]Kalita, J. C., and Sen, S., The (9,5) HOC formulation for the transient Navier-Stokes equations in primitive variable, Int. J. Numer. Meth. Fluids., 55 (2007), 387–406.Google Scholar
[7]Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, Second edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004.Google Scholar
[8]Strikwerda, J. C., and Lee, Y. S., The accuracy of the fractional step method, SIAM J. Numer. Anal., 37(1) (1999), 37–47.Google Scholar
[9]Strikwerda, J. C., High-order-accurate schemes for incompressible viscous flow, Int. J. Numer. Methods. Fluids., 24(7) (1997), 715–734.3.0.CO;2-E>CrossRefGoogle Scholar
[10]E, W.-N., and Liu, J.-G., Projection method, I: convergence and numerical boundary layers, SIAM J. Numer. Anal., 32(4) (1995), 1017–1057.Google Scholar
[11]E, W.-N., and Liu, J.-G., Projection method, II: Godunov-Ryabenki analysis, SIAM J. Numer. Anal., 33(4) (1996), 1597–1621.CrossRefGoogle Scholar
[12] W.-N. E, and Liu, J.-G., Projection method, III: spatial discretization on the staggered grid, Math. Comput., 71(237) (2002), 27–47.Google Scholar
[13]Brown, D. L., Ricardo, C., and Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168(2) (2001), 464–499.Google Scholar
[14]Karniadakis, G. Em., Israeli, M., and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97(2) (1991), 414–443.Google Scholar
[15]Johnston, H., and Liu, J.-G., Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199(1) (2004), 221–259.Google Scholar
[16]Wang, C., Liu, J.-G., and Johnston, H., Analysis of a fourth order finite difference method for the incompressible Boussinesq equations, Numer. Math., 97(3) (2004), 555–594.Google Scholar
[17]Liu, J.-G., Wang, C., and Johnston, H., A fourth order scheme for incompressible Boussinesq equations, J. Sci. Comput., 18(2) (2003), 253–285.Google Scholar
[18]Johnston, H., and Liu, J.-G., Finite difference schemes for incompressible flow based on local pressure boundary conditions, J. Comput. Phys., 180(1) (2002), 120–154.Google Scholar
[19]E, W.-N., and Liu, J.-G., Essentially compact schemes for unsteady viscous incompressible flows, J. Comput. Phys., 126(1) (1996), 122–138.CrossRefGoogle Scholar
[20]Ben-Artzi, M., Croisille, J.-P., Fishelov, D., and Trachtenberg, S., A pure-compact scheme for the streamfunction formulation of Navier-Stokes equations, J. Comput. Phys., 205 (2005), 640–664.CrossRefGoogle Scholar
[21]Ben-Artzi, M., Croisille, J.-P., and Fishelov, D., Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier-Stokes system, SIAM J. Numer. Anal., 44(5) (2006), 1997–2024.Google Scholar
[22]Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, Revised edition, with an appendix by Thomasset, F., Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.Google Scholar
[23]Orszag, S. A., Israeli, M., and Deville, M. O., Boundary conditions for incompressible flows, J. Sci. Comput., 1 (1986), 75–111.CrossRefGoogle Scholar
[24]Bell, J. B., Colella, P., and Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85(2) (1989), 257–283.CrossRefGoogle Scholar
[25]Van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput., 7(3) (1986), 870–891.CrossRefGoogle Scholar
[26]Gustafsson, B., The convergence rate for difference approximations to general mixed initial-boundary value problems, SIAM J. Numer. Anal., 18(2) (1981), 179–190.Google Scholar
[27]Gustafsson, B., Kreiss, H.-O., and Sundström, A., Stability theory of difference approximations for mixed initial-boundary value problems, II, Math. Comput., 26 (1972), 649–686.CrossRefGoogle Scholar
[28]Osher, S., Stability of difference equations of dissipative type for mixed initial-boundary value problems, I, Math. Comput., 23 (1969), 335–340.Google Scholar
[29]Kreiss, H. O., and Lorenz, J., Initial-Boundary Value Problems and the Navier-Stokes Equations, Pure and Applied Mathematics, 136, Academic Press, Inc., Boston, MA, 1989.Google Scholar
[30]Beam, R. M., and Warming, R. F., Alternating direction implicit methods for parabolic equations with a mixed derivative, SIAM J. Sci. Statist. Comput., 1(1) (1980), 131–159.CrossRefGoogle Scholar
[31]Mitchell, A. R., and Fairweather, G., Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for solving parabolic and elliptic equations, Numer. Math., 6 (1964), 285–292.Google Scholar
[32]Guermond, J. L., Minev, P., and Shen, J., An overview of projection methods for incompressible flows, Comput. Methods. Appl. Mech. Engrg., 195(44-47) (2006), 6011–6045.Google Scholar
[33]Henshaw, W. D., A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids, J. Comput. Phys., 113(1) (1994), 13–25.Google Scholar
[34]Henshaw, W. D., Kreiss, H.-O, and Reyna, L. G. M., A fourth-order-accurate difference approximation for the incompressible Navier-Stokes equations, Comput. Fluids., 23(4) (1994), 575–593.CrossRefGoogle Scholar
[35]Yanenko, N. N., The Method of Fractional Steps, The Solution of Problems of Mathematical Physics in Several Variables, Translated from the Russian by Cheron, T.. English translation edited by Holt, M., Springer-Verlag, New York-Heidelberg, 1971.Google Scholar
[36]Gustafsson, B., Kreiss, H.-O., and Oliger, J., Time Dependent Problems and Difference Methods, Wiley-Interscience, 1996.Google Scholar
[37]Pyo, J. H., and Shen, J., Normal mode analysis of second-order projection methods for incompressible flows, Discrete. Contin. Dyn. Syst. Ser. B., 5(3) (2005), 817–840.Google Scholar
[38]Peaceman, D. W., and Rachford, H. H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1955), 28–41.Google Scholar
[39]Ghia, U., Ghia, K. N., and Shin, C. T., High-re solutions for imcompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1987), 387–411.CrossRefGoogle Scholar
[40]Nishida, H., and Satofuka, H., Higher-order solutions of square driven cavity flow using a variable-order multi-grid method, Int. J. Numer. Methods. Eng., 34 (1992), 637–653.Google Scholar
[41]Armfield, S. W., Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids, Comput. Fluids., 20 (1991), 1–17.CrossRefGoogle Scholar