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Analysis of a fourth order finite difference method for the incompressible Boussinesq equations

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Summary.

The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at ‘‘ghost’’ points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.

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References

  1. Bathe, K.J. (ed.): First MIT Meeting on Computational Solid and Fluid Mechanics, vols. 1 and 2. MIT Elsevier: New York, 2001

  2. Bell, J.B., Marcus, D.L.: A Second-order projection method for variable-density flows. Journal of Computational Physics 101, 334–348 (1992)

    Google Scholar 

  3. Ben-Artzi, M., Fishelov, D., Trachtenberg, S.: Vorticity dynamics and numerical resolution of Navier-Stokes equations. Mathematical Modelling and Numerical Analysis 35, 313–330 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Briley, W.R.: A Numerical study of laminar separation bubbles using the Navier-Stokes equations. Journal of Fluid Mechanics 47, 713–736 (1971)

    MATH  Google Scholar 

  5. Carpenter, M.H., Gottlieb, D., Abarbanel, S.: The stability of numerical boundary treatments for compact high-order finite-difference schemes. Journal of Computational Physics 108, 272–295 (1993)

    Google Scholar 

  6. Christon, M., Gresho, P., Sutton S.B.: Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution). International Journal of Numerical Methods in Fluids 40, 953–980 (2001)

    Article  MATH  Google Scholar 

  7. E, W., Liu, J.-G.: Vorticity boundary condition for finite difference schemes. Journal of Computational Physics 124, 368–382 (1996)

    Google Scholar 

  8. E, W., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. Journal of Computational Physics 126, 122–138 (1996)

    Google Scholar 

  9. Glowinski, R., Pironneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review 21, 167–212 (1979)

    MathSciNet  MATH  Google Scholar 

  10. Henshaw, W.D., Kreiss, H.O., Reyna, L.G.M.: A Fourth-order accurate difference approximation for the incompressible Navier-Stokes equations. Computers and Fluids 23, 575–593 (1994)

    Google Scholar 

  11. Hou, T.Y., Wetton, B.: Convergence of a finite difference scheme for the Navier-Stokes equations using vorticity boundary conditions. SIAM Journal of Numerical Analysis 29, 615–639 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Hou, T.Y., Wetton, B.: Stable fourth order stream-function methods for incompressible flows with boundaries. Unpublished (1992)

  13. Johnston, H., Krasny, R.: Fourth order finite difference simulation of a differentially-heated cavity. International Journal of Numerical Methods in Fluids 40, 1031–1037 (2002)

    Article  MATH  Google Scholar 

  14. Johnston, H., Liu, J.-G.: Finite difference schemes for incompressible flow based on local pressure boundary conditions. Journal of Computational Physics 80, 120–154 (2002)

    Google Scholar 

  15. Kreiss, H.O.: Comparison of accurate methods for the integration of hyperbolic equations, Tellus, XXIV 3, 199–215 (1972)

  16. Liu, J.-G., Wang, C., Johnston, H.: A Fourth order scheme for incompressible Boussinesq equations. Journal of Scientific Computing 18, 253–285 (2003)

    Google Scholar 

  17. Orszag, S.A., Israeli, M.: Numerical simulation of viscous incompressible flows. Annual Review of Fluid Mechanics 6, 281–318 (1974)

    Article  MATH  Google Scholar 

  18. Quartapelle, L.: Numerical solution of the incompressible Navier-Stokes equations. Birkhauser, Berlin: 1983

  19. Spotz, W.F., Carey, G.F.: High-order compact scheme for the stream-function vorticity equations. International Journal for Numerical Methods in Engineering 38 (20), 3497–3512 (1995)

    MATH  Google Scholar 

  20. Thom, A.: The flow past circular cylinders at low speeds. Proceedings Royal Society A141, 651–669 (1933)

  21. Wang, C., Liu, J.-G.: Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. Numerische Mathematik 91, 543–576 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

    Cheng Wang

  2. Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, MD 20742, USA

    Jian-Guo Liu

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

    Hangs Johnston

Authors
  1. Cheng Wang
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  2. Jian-Guo Liu
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  3. Hangs Johnston
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Correspondence to Cheng Wang.

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Mathematics Subject Classification (1991): 35Q35, 65M06, 76M20

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Wang, C., Liu, JG. & Johnston, H. Analysis of a fourth order finite difference method for the incompressible Boussinesq equations. Numer. Math. 97, 555–594 (2004). https://doi.org/10.1007/s00211-003-0508-3

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  • DOI: https://doi.org/10.1007/s00211-003-0508-3

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