Advertisement

Aerodynamic applications of Newton- Krylov-Schwarz solvers

  • David E. Keyes
1. Invited Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 453)

Abstract

Parallel implicit solution methods are increasingly important in aerodynamics, since reliable low-residual solutions at elevated CFL number are prerequisite to such large-scale applications of aerodynamic analysis codes as aeroelasticity and optimization. In this chapter, a class of nonlinear implicit methods and a class of linear implicit methods are defined and illustrated. Their composition forms a class of methods with strong potential for parallel implicit solution of aerodynamics problems. Newton-Krylov methods are suited for nonlinear problems in which it is unreasonable to compute or store a true Jacobian, given a strong enough preconditioner for the inner linear system that needs to be solved for each Newton correction. In turn, Krylov-Schwarz iterative methods are suited for the parallel implicit solution of multidimensional systems of linearized boundary value problems. Schwarz-type domain decomposition preconditioning provides good data locality for parallel implementations over a range of granularities. These methods are reviewed separately, illustrated with CFD applications, and composed in a class of methods named Newton-Krylov-Schwarz.

Keywords

Coarse Grid Domain Decomposition Unstructured Grid Implicit Method Domain Decomposition Method 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W.K. Anderson and D.L. Bonhaus (1994): An Implicit upwind Algorithm for Computing Turbulent Flows on Unstructured Grids, Computers & Fluids 23 1–21.Google Scholar
  2. 2.
    O. Axelsson and V.A. Barker (1983): Finite Element Solution of Boundary Value Problems: Theory and Computation, Academic Press, New York.Google Scholar
  3. 3.
    P.N. Brown and A. Hindmarsh (1987): Matrix-free Methods for Stiff Systems of ODES, SIAM J. Numer. Anal. 24 610–638.CrossRefGoogle Scholar
  4. 4.
    P.N. Brown and Y. Saad (1990): Hybrid Krylov Methods for Nonlinear Systems of Equations, SIAM J. Sci. Stat. Comp. 11 450–481.CrossRefGoogle Scholar
  5. 5.
    X.-C. Cai, W.D. Gropp, D.E. Keyes and M.D. Tidriri (1994): Newton-KrylovSchwarz in CFD, in “Proceedings of the International Workshop on Numerical Methods for the Navier-Stokes Equations” (F. Hebeker & R. Rannacher, eds.), Notes in Numerical Fluid Mechanics, Vieweg Verlag, Braunschweig, pp. 17–30.Google Scholar
  6. 6.
    T.F. Chan, R. Glowinski, J. Periaux, and O.B. Widlund, eds. (1989): Proc. of the Second Intl. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia.Google Scholar
  7. 7.
    T.F. Chan, R. Glowinski, J. Periaux, and O.B. Widlund, eds. (1990): Proc. of the Third Intl. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia.Google Scholar
  8. 8.
    T.F. Chan and K.R. Jackson (1986): The Use of Iterative Linear Equation Solvers in Codes for Large Stiff System of IVPs for ODEs, SIAM J. Sci. Stat. Comp. 7 378–417.CrossRefGoogle Scholar
  9. 9.
    M. Y.-M. Chang and M.H. Schultz (1994): Bounds on Block Diagonal Preconditioning, Parallel Algs. and Applics. 1 141–164.Google Scholar
  10. 10.
    J.G. Chefter, C.K. Chu and D.E. Keyes (1995): Domain Decomposition for the Shallow Water Equations, in “Proceedings of the Seventh International Conference on Domain Decomposition Methods” (D.E. Keyes & J. Xu, eds.), AMS, Providence (to appear).Google Scholar
  11. 11.
    R. Dembo, S. Eisenstat and T. Steihaug (1982): Inexact Newton Methods, SIAM J. Numer. Anal. 19 400–408.CrossRefGoogle Scholar
  12. 12.
    J.E. Dennia, Jr. and R.B. Schnabel (1983): Numerical Methods for Unconstrained Optimization & Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  13. 13.
    M. Dryja and O.B. Widlund (1987): An Additive Variant of the Alternatively Method for the Case of Many Subregions, TR 339, Courant Institute, NYU.Google Scholar
  14. 14.
    A. Ern, V. Giovangigli, D.E. Keyes and M.D. Smooke (1994): Towards Polyalgoriihmic Linear System Solvers for Nonlinear Elliptic Problems, SIAM J. Sci. Comp. 15 681–703.CrossRefGoogle Scholar
  15. 15.
    C. Farhat, L. Crivelli and F.-X. Roux, extending Substructure-based Iterative Solvers to Multiple Load and Repeated Analyses, Comput. Meths. Appl. Mech. Engrg. *37 195–209.Google Scholar
  16. 16.
    C. Farhat, S. Lanteri and H.D. Simon (1994): TOP/DOMDEC: A Software Tool for Mesh Partitioning and Parallel Processing, J. Comput. Sys. Engrg. (to appear).Google Scholar
  17. 17.
    P.F. Fischer (1993): Projection Techniques for Iterative Solution of Ax = b with Successive Right-hand sides, Technical Report 93-90, ICASE, NASA Langley Res. Ctr.Google Scholar
  18. 18.
    R.W. Freund and N.M. Nachtigal (1991) QMR: A Quali-minimal Residual Methods for Non-Hermitian Linear Systems, Numer. Math. 60 315–339.CrossRefGoogle Scholar
  19. 19.
    R. Glowinski, G.H. Golub, G.A. Meurant, and J. Periaux, eds. (1988): Proc. of the First Intl. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia.Google Scholar
  20. 20.
    R. Glowinski, Yu. A. Kuznetsov, G.A. Meurant, and O.B. Widlund, eds. (1991): Proc. of the Fourth Intl Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM. Phiadelphia.Google Scholar
  21. 21.
    G.H. Golub and C.F. van Loan (1989): Matrix Computations, Johns Hopkins, Baltimore.Google Scholar
  22. 22.
    Z. Johann, T.J.R. Hughes and F. Shakib (1991): A Globally Convergent Matrix-Free Algorithm for Implicit Time-Marching Schemes Arising in Finite Element Analysis in Fluids, Comp. Meths. Appl. Mech. Engrg. 87 281–304.CrossRefGoogle Scholar
  23. 23.
    D.E. Keyes, T.F. Chan, G.A. Meurant, J.S. Scroggs, and R.G. Voigt, eds. (1992): Proc. of the Fifth Intl. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia.Google Scholar
  24. 24.
    D.E. Keyes, Y. Saad and D.G. Truhlar eds. (1995): Domain-based Parallelism and Problem Decomposition Methods in Science and Engineering, SIAM, Philadelphia (to appear).Google Scholar
  25. 25.
    D.E. Keyes and J. Xu, eds. (1995): Proc. of the Seventh Intl. Symp. on Domain Decomposition Methods in Science and Engineering, AMS, Providence (to appear).Google Scholar
  26. 26.
    D.A. Knoll and P.R. McHugh (1993): Inexact Newton's Method Solutions to the Incompressible Navier-Stokes and Energy Equations Using Standard and Matrix-Free Implementations, AIAA Paper 93-3332.Google Scholar
  27. 27.
    E.J. Nielsen, R.W. Walters, W.K. Anderson and D.E. Keyes (1995): Application of Newton-Krylov Methodology to a Three-Dimensional Unstructured Euler Code, in “Proceedings of the 12th AIAA Computational Fluid Dynamics Conference” (to appear).Google Scholar
  28. 28.
    K.G. Prasad, D.E. Keyes and J.H. Kane (1994): GMRES for Sequentially Multiple Right-hand Sides, SIAM J. Sci. Comp. (submitted).Google Scholar
  29. 29.
    A. Quarteroni, J. Periaux, Yu. A. Kuznetsov, and O.B. Widlund, eds. (1994): Proc. of the Sixth Intl. Symp. on Domain Decomposition Methods in Science and Engineering, AMS, Providence.Google Scholar
  30. 30.
    Y. Saad and M.H. Schultz (1986): GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmeiric Linear Systems, SIAM J. Sci. Stat. Comp. 7 865–869.CrossRefGoogle Scholar
  31. 31.
    J.L. Steger and R.F. Warming (1981): Flux Vector Splitting of the Inviscid Gasdynamic Equations with Applications of Finite-Difference Methods, J. Comp. Phys. 40 263–293.CrossRefGoogle Scholar
  32. 32.
    E. Turkel, A. Fiterman and B. van Leer (1993): Preconditioning and the Limit to the Incompressible Flow Equations, Technical Report 93-42, ICASE, NASA Langley Res. Ctr.Google Scholar
  33. 33.
    H.A. Van der Vorst (1992): Bi-CGSTAB: A More Smoothly Converging Variant of CG-S for the Solution of Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comp. 13 631–644.CrossRefGoogle Scholar
  34. 34.
    L.B. Wigton, N.J. Yu and D.P. Young (1985): GMRES Acceleration of Computational Fluid Dynamics Codes, AIAA Paper 85-1494.Google Scholar
  35. 35.
    J. Xu (1992): Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34 581–613.CrossRefGoogle Scholar
  36. 36.
    D.P. Young, C.C. Ashcraft, R.G. Melvin, M.B. Bieterman, W.P. Huffman, T.F. Johnson, C.L. Hilmes, and J.E. Bussoletti (1993): Ordering and Incomplete Factorization Issues for Matrices Arising from the TRANAIR CFD Code, TR BCSTECH-93-025, Boeing Computer Services, Seattle.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • David E. Keyes
    • 1
    • 2
  1. 1.Institute for Computer Applications in Science and Engg.NASA Langley Research CenterHamptonUSA
  2. 2.Department of Computer ScienceOld Dominion UniversityNewfolkUSA

Personalised recommendations