Advertisement

Exploring Parallel-in-Time Approaches for Eddy Current Problems

  • Stephanie FriedhoffEmail author
  • Jens Hahne
  • Iryna Kulchytska-Ruchka
  • Sebastian Schöps
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)

Abstract

We consider the usage of parallel-in-time algorithms of the Parareal and multigrid-reduction-in-time (MGRIT) methodologies for the parallel-in-time solution of the eddy current problem. Via application of these methods to a two-dimensional model problem for a coaxial cable model, we show that a significant speedup can be achieved in comparison to sequential time stepping.

Notes

Acknowledgements

The work is supported by the Excellence Initiative of the German Federal and State Governments, the Graduate School of Computational Engineering at TU Darmstadt, and the BMBF in the framework of project PASIROM (grants 05M18RDA and 05M18PXB).

References

  1. 1.
    Nievergelt, J.: Parallel methods for integrating ordinary differential equations. Commun. Assoc. Comput. Mach. 7, 731–733 (1964)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition, pp. 69–113. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  3. 3.
    Lions, J.-L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDEs. C. R. Acad. Sci. 332, 661–668 (2001)CrossRefGoogle Scholar
  4. 4.
    Falgout, R.D., Friedhoff, S., Kolev, T. V., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36(6), C635–C661 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Meeker, D.C.: Finite Element Method Magnetics, Version 4.2 (28 Feb 2018 Build). http://www.femm.info
  6. 6.
    Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998)zbMATHGoogle Scholar
  7. 7.
    Heise, B.: Analysis of a fully discrete finite element method for a nonlinear magnetic field problem. SIAM J. Numer. Anal. 31(3), 745–759 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Schmidt, K., Sterz, O., Hiptmair, R.: Estimating the eddy-current modeling error. IEEE Trans. Magn. 44(6), 686–689 (2008)CrossRefGoogle Scholar
  9. 9.
    Emson, C.R.I., Trowbridge, C.W.: Transient 3d eddy currents using modified magnetic vector potentials and magnetic scalar potentials. IEEE Trans. Magn. 24(1), 86–89 (1988)CrossRefGoogle Scholar
  10. 10.
    Nicolet, A., Delincé, F.: Implicit Runge-Kutta methods for transient magnetic field computation. IEEE Trans. Magn. 32(3), 1405–1408 (1996)CrossRefGoogle Scholar
  11. 11.
    Ries, M., Trottenberg, U.: MGR-Ein blitzschneller elliptischer Löser. Preprint 277 SFB 72. Universität Bonn, Bonn (1979)Google Scholar
  12. 12.
    Ries, M., Trottenberg, U., Winter, G.: A note on MGR methods. Linear Algebra Appl. 49, 1–26 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gander, M.J., Kulchytska-Ruchka, I., Niyonzima, I., Schöps, S.: A New Parareal Algorithm for Problems with Discontinuous Sources. Submitted to SISC, arXiv: 1803.05503 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Stephanie Friedhoff
    • 1
    Email author
  • Jens Hahne
    • 1
  • Iryna Kulchytska-Ruchka
    • 2
  • Sebastian Schöps
    • 2
  1. 1.Fakultät für Mathematik und NaturwissenschaftenBergische Universität WuppertalWuppertalGermany
  2. 2.Centre for Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations