The Efficient Parallel Newton-GMRES Algorithm for Computational Fluid Dynamics

  • Tianruo Yang
Conference paper
Part of the Astrophysics and Space Science Library book series (ASSL, volume 263)

Abstract

The main motivation to solve the compressible Navier-Stokes equations by means of an implicit method is the excessive small time steps limited by stability constraints for explicit methods. Thus, for structured meshes, work has been done both on the development of new implicit schemes [13] and on the use of new tools to solve large sparse unsymmetric linear systems [10, 19]. Also for unstructured meshes, new schemes [11, 18] and techniques [12] have been developed. However, in most cases, each implicit step leads to one costly nonlinear system to be solved.

Keywords

Computational Fluid Dynamics Krylov Subspace GMRES Method Nonsymmetric Linear System Logical Ring 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Z. Bai, D. Hu, and L. Reichel. A newton basis GMRES implementation. Technical Report 91-03, University of Kentucky, 1991.Google Scholar
  2. 2.
    P. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM Journal on Scientific and Statistical Computing, 11:450-481, 1990.CrossRefGoogle Scholar
  3. 3.
    A. T. Chronopoulos and S. K. Kim. s-step OTHEROMIN and GMRES implemented on parallel computers. Technical Report 90-R-43, Supercomputer Institute, University of Minnesota, 1990.Google Scholar
  4. 4.
    L. G. C. Crone and H. A. van der Vorst. Communication aspects of the conjugate gradient method on distributed memory machines. Supercomputer, X(6):4-9, 1993.Google Scholar
  5. 5.
    E. de Sturler. A parallel variant of the GMRES(m). In Proceedings of the 13th IMACS World Congress on Computational and Applied Mathematics. IMACS, Criterion Press, 1991.Google Scholar
  6. 6.
    E. de Sturler and H. A. van der Vorst. Reducing the effect of the global communication in GMRES(m) and CG on parallel distributed memory computers. Technical Report 832, Mathematical Institute, University of Utrecht, Utrecht, The Netheland, 1994.Google Scholar
  7. 7.
    J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst. Solving Linear Systems on Vector and Shared Memory Computers. SIAM, Philadelphia, PA, 1991.Google Scholar
  8. 8.
    J. Erhel. A parallel GMRES version for general sparse matrices. Electronic Transactions on Numerical Analysis, 3:160-176, 1995.Google Scholar
  9. 9.
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, 1991.Google Scholar
  10. 10.
    R. Hixon, F.L. Tsung, and L. N. Sankar. A comparison of two methods for solving 3D unsteady compressible viscous flows. In Proceedings of the 31th Aerospace Science Meeting. AAAI, 1993.Google Scholar
  11. 11.
    T. J.R. Hughes, L. P. Franca, and M. Mallet. A new finite element formulation for computational fluid dynamics. Computer Methods for Applied Mechanics and Engineering, 54:223-234, 1986.CrossRefGoogle Scholar
  12. 12.
    Z. Johan, T. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time-searching schemes arising in finite element analysis in fluids. Computer Methods in Applied Mechanics and Engineering, 87:281-304, 1991.CrossRefGoogle Scholar
  13. 13.
    A Lerat and J. Sides. Efficient solution of the steady euler equations with a centered implicit method. In K. W. Morton and M. J. Baines, editors, Numerical and Mechanics Fluid Dynamics, volume III, pages 65-86. 1988.Google Scholar
  14. 14.
    C. Pommerell. Solution of large unsymmetric systems of linear equations. PhD thesis, ETH, 1992.Google Scholar
  15. 15.
    Y. Saad. Krylov subspace methods in distributed computing enviroments. Technical Report TR-92-126, Army High Performance Computing Center, Minneapolis, 1992.Google Scholar
  16. 16.
    Y. Saad and A. Malvesky. P-Sparslib: a portable library of distributed memory sparse iterative solvers. Technical Report UMSI-95-180, University of Minnesota, 1995.Google Scholar
  17. 17.
    Y. Saad and M. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7:856-869, 1986.CrossRefGoogle Scholar
  18. 18.
    A. Soulaimani and M. Fortin. Finite element solution of compressible visons flows using conservative variables. Computer Methods for Applied Mechanics and Engineering, 118:319-350, 1994.CrossRefGoogle Scholar
  19. 19.
    L. B. Wigton, N. J. Yu, and D. P. Young. GMRES acceleration of computational fluid dynamics codes. In Proceedings of the 7th Computational Fluid Dynamics. AAAI, 1985.Google Scholar
  20. 20. T. Yang. Solving sparse least squares problems on massively parallel distributed memory computers. In Proceedings of International Conference on Advances in Parallel and Distributed Computing (APDC-97), March 1997. Shanghai, P.R.China.Google Scholar
  21. 21. T. Yang and II. X. Lin. The improved quasi-minimal residual method on massively distributed memory computers. In Proceedings of The International Conference on High Performance Computing and Networking (HPCN-97), April 1997.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Tianruo Yang
    • 1
  1. 1.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden

Personalised recommendations