Advertisement

Krylov Smoothing for Fully-Coupled AMG Preconditioners for VMS Resistive MHD

  • Paul T. LinEmail author
  • John N. Shadid
  • Paul H. Tsuji
Chapter
  • 98 Downloads
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 132)

Abstract

This study explores the use of a Krylov iterative method (GMRES) as a smoother for an algebraic multigrid (AMG) preconditioned Newton–Krylov iterative solution approach for a fully-implicit variational multiscale (VMS) finite element (FE) resistive magnetohydrodynamics (MHD) formulation. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play an essential role. Krylov smoothers are considered an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. This brief study presents three time dependent resistive MHD test cases to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.

Keywords

Multigrid Krylov smoother Preconditioner Newton–Krylov Finite element method Magnetohydrodynamics 

Notes

Acknowledgements

The authors would like to thank R. Pawlowski and E. Cyr for their collaborative effort in developing the Drekar MHD code and D. Sondak and T. Smith for the collaborative development of the VMS LES turbulent MHD modeling capability. The authors gratefully acknowledge the generous support of the DOE NNSA ASC program and the DOE Office of Science AMR program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy, National Nuclear Security Administration under contract DE-NA-0003525.

References

  1. 1.
    Bank, R.E., Douglas, C.C.: Sharp estimates for multigrid rates of convergence with general smoothing and acceleration. SIAM J. Numer. Anal. 22, 617–633 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Birken, P., Bull, J., Jameson, A.: A study of multigrid smoothers used in compressible CFD based on the convection diffusion equations. In: Papadrakakis M., et al. (eds.) Proceedings of the VII ECCOMAS Congress 2016, Crete Island (2016)Google Scholar
  3. 3.
    Bornemann, F.A., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numerische Mathematik 75, 135–152 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Braess, D.: On the combination of the multigrid method and conjugate gradients. In: Multigrid Methods II. Lecture Notes in Mathematics, vol. 1228, pp. 52–64. Springer, Berlin (2006)Google Scholar
  5. 5.
    Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450–481 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cai, X.-C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chacón, L.: A non-staggered, conservative, ∇⋅B = 0, finite-volume scheme for 3D implicit extended magnetohydrodynamics in curvilinear geometries. Comput. Phys. Comm. 163, 143–171 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chacón, L.: An optimal, parallel, fully implicit Newton-Krylov solver for three-dimensional visco-resistive magnetohydrodynamics. Phys. Plasmas 15, 056103 (2008)CrossRefGoogle Scholar
  9. 9.
    Codina, R., Hernández-Silva, N.: Stabilized finite element approximation of the stationary magneto-hydrodynamics equations. Comput. Mech. 38, 344–355 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Elman, H.C., Ernst, O.G., O’Leary, D.P.: A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J. Sci. Commun. 23, 1291–1315 (2001)CrossRefGoogle Scholar
  11. 11.
    Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649. Sandia National Laboratories, New Mexico (2006)Google Scholar
  12. 12.
    Goedbloed, H., Poedts, S.: Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  13. 13.
    Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lin, P.T., Sala, M., Shadid, J.N., Tuminaro, R.S.: Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. Int. J. Num. Meth. Eng. 67(2), 208–225 (2006)CrossRefGoogle Scholar
  15. 15.
    Lin, P., Shadid, J., Tsuji, P., Hu, J.J.: Performance of smoothers for algebraic multigrid preconditioners for finite element variational multiscale incompressible magnetohydrodynamics. In: Proceedings of SIAM PP16. SIAM, Philadelphia (2016)Google Scholar
  16. 16.
    Notay, Y., Vassilevski, P.: Recursive Krylov-based multigrid cycles. Numer. Linear Algebra Appl. 15, 473–487 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  18. 18.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  19. 19.
    Shadid, J.N., Tuminaro, R.S., Devine, K.D., Hennigan, G.L., Lin, P.T.: Performance of fully-coupled domain decomposition preconditioners for finite element transport/reaction simulations. J. Comput. Phys. 205(1), 24–47 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shadid, J.N., Pawlowski, R.P., Banks, J.W., Chacón, L., Lin, P.T., Tuminaro, R.S.: Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. J. Comput. Phys. 229(20), 7649–7671 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Shadid, J.N., Pawlowski, R.P., Cyr, E.C., Tuminaro, R.S., Chacon, L., Weber, P.D.: Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov–AMG. Comput. Methods Appl. Mech. Eng. 304, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sondak, D., Shadid, J.N., Oberai, A.A., Pawlowski, R.P., Cyr, E.C., Smith, T.M.: A new class of finite element variational multiscale turbulence models for incompressible magnetohydrodynamics. J. Comput. Phys. 295, 596–616 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic, London (2001)zbMATHGoogle Scholar
  24. 24.
    Tuminaro, R.S., Heroux, M., Hutchinson, S.A., Shadid, J.N.: Aztec User’s Guide–Version 2.1. Technical Report SAND99-8801J. Sandia National Laboratories, New Mexico (1999)Google Scholar
  25. 25.
    Van der Vorst, H.A., Vuik, C.: GMRESR: a family of nested GMRES methods. Numer. Linear Algebra Appl. 1, 369–386 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA
  2. 2.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA
  3. 3.Lawrence Livermore National LaboratoryLivermoreUSA

Personalised recommendations