The Role of Multi-method Linear Solvers in PDE-based Simulations

  • S. Bhowmick
  • L. McInnes
  • B. Norris
  • P. Raghavan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2667)


The solution of large-scale, nonlinear PDE-based simulations typically depends on the performance of sparse linear solvers, which may be invoked at each nonlinear iteration. We present a framework for using multi-method solvers in such simulations to potentially improve the execution time and reliability of linear system solution. We consider composite solvers, which provide reliable linear solution by using a sequence of preconditioned iterative methods on a given system until convergence is achieved.We also consider adaptive solvers, where the solution method is selected dynamically to match the attributes of linear systems as they change during the course of the nonlinear iterations.We demonstrate how such multi-method composite and adaptive solvers can be developed using advanced software architectures such as PETSc, and we report on their performance in a computational fluid dynamics application.


Linear Solver Linear Solution Residual Norm Inexact Newton Method Drive Cavity 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CCA1]
    R. Armstrong, D. Gannon, A. Geist, K. Keahey, S. Kohn, L. C. McInnes, S. Parker, and B. Smolinski, Toward a Common Component Architecture for High-Performance Scientific Computing, Proceedings of High Performance Distributed Computing (1999) pp. 115–124, (see
  2. [A1]
    O. Axelsson, A survey of preconditioned iterative methods for linear systems of equations, BIT, 25 (1987) pp. 166–187.MathSciNetGoogle Scholar
  3. [P1]
    S. Balay, K. Buschelman, W. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B. Smith, and H. Zhang, PETSc users manual, Tech. Rep. ANL-95/11-Revision 2.1.3, Argonne National Laboratory, 2002 (see
  4. [P2]
    S. Balay, W. Gropp, L.C. McInnes, and B. Smith, Efficient management of parallelism in object oriented numerical software libraries, In Modern Software Tools in Scientific Computing (1997), E. Arge, A. M. Bruaset, and H. P. Langtangen, Eds., Birkhauser Press, pp. 163–202.Google Scholar
  5. [B1]
    R. Barrett, M. Berry, J. Dongarra, V. Eijkhout, and C. Romine, Algorithmic Bombardment for the Iterative Solution of Linear Systems: A PolyIterative Approach. Journal of Computational and applied Mathematics, 74, (1996) pp. 91–110.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [MM1]
    S. Bhowmick, P. Raghavan, and K. Teranishi, A Combinatorial Scheme for Developing Efficient Composite Solvers, Lecture Notes in Computer Science, Eds. P. M. A. Sloot, C. J. K. Tan, J. J. Dongarra, A. G. Hoekstra, 2330, Springer Verlag, Computational Science-ICCS 2002, (2002) pp. 325–334.Google Scholar
  7. [MM2]
    S. Bhowmick, P. Raghavan, L. McInnes, and B. Norris, Faster PDE-based simulations using robust composite linear solvers. Argonne National Laboratory preprint ANL/MCS-P993-0902, 2002. Also under review, Future Generation Computer SystemsGoogle Scholar
  8. [LSA1]
    R. Bramley, D. Gannon, T. Stuckey, J. Villacis, J. Balasubramanian, E. Akman, F. Berg, S. Diwan, and M. Govindaraju, The Linear System Analyzer, Enabling Technologies for Computational Science, Kluwer, (2000).Google Scholar
  9. [CKK1]
    T. S. Coffey, C. T. Kelley, and D. E. Keyes, Pseudo-Transient Continuation and Differential-Algebraic Equations, submitted to the open literature, 2002.Google Scholar
  10. [DEK1]
    I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, Clarendon Press, Oxford, 1986.zbMATHGoogle Scholar
  11. [EGKS1]
    A. Ern, V. Giovangigli, D. E. Keyes, and M. D. Smooke, Towards Polyalgorithmic Linear System Solvers For Nonlinear Elliptic Problems, SIAM J. Sci. Comput., Vol. 15, No. 3, pp. 681–703.Google Scholar
  12. [FGN1]
    R. Freund, G. H. Golub, and N. Nachtigal, Iterative Solution of Linear Systems, Acta Numerica, Cambridge University Press, (1992) pp. 57–100.Google Scholar
  13. [FL1]
    B. Fryxell, K. Olson, P. Ricker, F. Timmes, M. Zingale, D. Lamb, P. MacNeice, R. Rosner, J. Truran, and H. Tufo, FLASH: an Adaptive-Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes, Astrophysical Journal Supplement, 131 (2000) pp. 273–334, (see Scholar
  14. [KK1]
    C. T. Kelley and D. E. Keyes, Convergence analysis of pseudo-transient continuation. SIAM Journal on Numerical Analysis 1998; 35:508–523.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [MM3]
    L. McInnes, B. Norris, S. Bhowmick, and P. Raghavan, Adaptive Sparse Linear Solvers for Implicit CFD Using Newton-Krylov Algorithms, To appear in the Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, Massachusetts Institute of Technology, Boston, USA, June 17–20, 2003.Google Scholar
  16. [NW1]
    J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.zbMATHGoogle Scholar
  17. [N1]
    B. Norris, S. Balay, S. Benson, L. Freitag, P. Hovland, L. McInnes, and B. F. Smith, Parallel Components for PDEs and Optimization: Some Issues and Experiences, Parallel Computing, 28(12) (2002), pp. 1811–1831.zbMATHCrossRefGoogle Scholar
  18. [T1]
    X. Z. Tang, G. Y. Fu, S. C. Jardin, L. L. Lowe, W. Park, and H. R. Strauss, Resistive Magnetohydrodynamics Simulation of Fusion Plasmas, PPPL-3532, Princeton Plasma Physics Laboratory, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • S. Bhowmick
    • 1
  • L. McInnes
    • 2
  • B. Norris
    • 2
  • P. Raghavan
    • 1
  1. 1.Department of Computer Science and EngineeringThe Pennsylvania State UniversityUniversity Park
  2. 2.Mathematics and Computer Sciences DivisionArgonne National LaboratoryArgonne

Personalised recommendations