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Parallelization of Solution Schemes for the Navier-Stokes Equations

  • J. Hofhaus
  • M. Meinke
  • E. Krause
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)

Abstract

An explicit and implicit scheme for the solution of the Navier-Stokes equations for unsteady and three-dimensional flow problems were implemented on several parallel computer systems. The explicit scheme is based on a multi-stage Runge-Kutta scheme with multigrid acceleration, in the implicit scheme a dual time stepping scheme and a conjugate gradient with incomplete lower-upper decomposition preconditioning is applied. Two examples of complex flows were simulated and compared with experimental flow visualizations in order to demonstrate the applicability of the developed solution methods. Presented are the essential details of the solution schemes, their implementation on parallel computer architectures, and their performance for different hardware configurations.

Keywords

Vortex Ring Coarse Grid Multigrid Method Implicit Scheme Explicit Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    B. Bartmann and R. Neikes. Experimentelle Untersuchung verzweigender Innenströmung II FKM/AIF-Vorhaben Nr.113, Abschlußbericht, 1991.Google Scholar
  2. [2]
    A. Brandt. Guide to multigrid development. In Lecture Notes in Mathematics, pages 220–312. Springer Verlag Berlin, 1981.Google Scholar
  3. [3]
    M. Breuer and D. Hänel. A dual time-stepping method for 3-d, viscous, incompressible vortex flows. Computers Fluids, 22(4/5):467–484, 1993.zbMATHCrossRefGoogle Scholar
  4. [4]
    Michael Breuer. Numerische Lösung der Navier-Stokes Gleichungen für dreidimensionale inkompressible instationäre Strömungen zur Simulation des Wirbelplatzens. Dissertation, Aerodynamisches Institut der RWTH-Aachen, Juni 1991.Google Scholar
  5. [5]
    A.J. Chorin. A Numerical Method for Solving Incompressible Viscous Flow. J. Comput. Phys., 2:12–26, 1967.zbMATHCrossRefGoogle Scholar
  6. [6]
    A. Harten. On a Class of High Resolution Total-Variation-Stable Finite Difference Schemes for Hyperbolic Conservation Laws. SIAM J. Numer. Anal., 21, 1984.Google Scholar
  7. [7]
    J. Hofhaus and M. Meinke. Parallel solution schemes for the navier-stokes equations. In S. Wagner, editor, Computational Fluid Dynamics on Parallel Systems, volume 50 of Notes on Numerical Fluid Mechanics, pages 88–96. Vieweg Verlag, 1995.Google Scholar
  8. [8]
    J. Hofhaus and E.F. Van De Velde. Alternating-Direction Line-Relaxation Methods on Multicomputers. SIAM J. Sci. Comput., 1996. in press.Google Scholar
  9. [9]
    E. Krause, M. Meinke, and J. Hofhaus. Experience with parallel computing in fluid mechanics. In ECCOMAS 94 John Wiley & Sons, Ltd., 1994.Google Scholar
  10. [10]
    B. P. Leonard. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19:59, 1979.zbMATHCrossRefGoogle Scholar
  11. [11]
    T. T. Lim. An Experimental Study of a Vortex Ring Interacting with an Inclined Wall. Experiments in Fluids, 7:453–463, 1989.CrossRefGoogle Scholar
  12. [12]
    M. Meinke and D. Hänel. Time accurate multigrid solutions of the Navier-Stokes equations. In W. Hackbusch and U. Trottenberg, editors, Multigrid Methods III, pages 289–300. Birkhäuser Verlag, 1991.Google Scholar
  13. [13]
    M. Meinke and E. Ortner. Implementation of Explicit Navier-Stokes Solvers on Massively Parallel Systems. In E. H. Hirschel, editor, DFG Priority Research Programme, Results 1989–1992, volume 38 of Notes on Numerical Fluid Mechanics, pages 138–151. Vieweg Verlag, 1993.Google Scholar
  14. [14]
    Matthias Meinke. Numerische Lösung der Navier-Stokes-Gleichung en für instationäre Strömungen mit Hilfe der Mehrgittermethode. Dissertation, Aerodynamisches Institut der RWTH-Aachen, März 1993.Google Scholar
  15. [15]
    Y. Nakamura and Y. Takemoto. Solutions of Incompressible Flows Using a Generalized QUICK Method. In Numerical Methods in Fluid Mechanics II, Proc. of the Int. Symp. on Comput. Fluid Dynamics, Tokyo, Sept. 9–12 1985.Google Scholar
  16. [16]
    P. L. Roe. Approximate riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43, 1981.Google Scholar
  17. [17]
    Y. Takemoto and Y. Nakamura. A three-dimensional incompressible flow solver. Lecture Notes in Physics, 264:594–599, 1986. Springer Verlag.CrossRefGoogle Scholar
  18. [18]
    H.A. Van Der Vorst. BI-CGSTAB: A Fast and Smoothly Converging Variant of BI-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comput., 13(2):631–644, 1992.zbMATHCrossRefGoogle Scholar
  19. [19]
    S. Yoon and D. Kwak. Three-Dimensional Incompressible Navier-Stokes Solver Using Lower-Upper Symmetric-Gauss-Seidel Algorithm. AIAA Journal, 29(6):874–875, 1991.CrossRefGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

Authors and Affiliations

  • J. Hofhaus
    • 1
  • M. Meinke
    • 1
  • E. Krause
    • 1
  1. 1.Aerodynamisches InstitutRWTH AachenAachenGermany

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