Numerische Mathematik

, Volume 129, Issue 3, pp 535–561 | Cite as

Convergence of an ADI splitting for Maxwell’s equations

  • Marlis Hochbruck
  • Tobias Jahnke
  • Roland Schnaubelt
Article

Abstract

The convergence of an alternating direction implicit method for Maxwell’s equations on product domains is investigated. Unlike the classical Yee scheme and most other integrators proposed in the literature, this method is both unconditionally stable and computationally cheap. We prove second-order convergence of the time-discretization in the framework of operator semigroup theory. In contrast to formal considerations based on Taylor expansions, our convergence analysis respects the unboundedness of the involved differential operators. The proofs are based on results concerning the regularity of the Cauchy problems, which then allow to apply an abstract convergence proof by Hansen and Ostermann (Numer Math 108:557–570, 2008).

Mathematics Subject Classification (2010)

Primary 65M12 Secondary 35Q61 47D03 65J10 

References

  1. 1.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823–864 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Chen, W., Li, X., Liang, D.: Energy-conserved splitting FDTD methods for Maxwell’s equations. Numer. Math. 108(3), 445–485 (2008)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151(3), 221–276 (2000)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology. In: Spectral Theory and Applications. With the Collaboration of Michel Artola and Michel Cessenat, Vol. 3, 2nd edn. Springer, Berlin (2000a)Google Scholar
  5. 5.
    Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology. In: Evolution Problems. I. With the Collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Vol. 5, 2nd edn. Springer, Berlin (2000b)Google Scholar
  6. 6.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics. Springer, Berlin (2000)Google Scholar
  7. 7.
    Faragó, I., Horváth, R., Schilders, W.H.A.: Investigation of numerical time-integrations of Maxwell’s equations using the staggered grid spatial discretization. Int. J. Numer. Model. Electron. Netw. Dev. Fields 18(2), 149–169 (2005)CrossRefMATHGoogle Scholar
  8. 8.
    Gao, L., Zhang, B., Liang, D.: The splitting finite-difference time-domain methods for Maxwell’s equations in two dimensions. J. Comput. Appl. Math. 205(1), 207–230 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Garcia, S.G., Lee, Tae-Woo, Hagness, S.C.: On the accuracy of the ADI-FDTD method. IEEE Antennas Wirel. Propag. Lett. 1(1), 31–34 (2002)CrossRefGoogle Scholar
  10. 10.
    Gonzáles García, S., Rubio Bretones, A., Gómez Martín, R., Hagness, S.C.: Accurate implementation of current sources in the ADI-FDTD scheme. IEEE Antennas Wirel. Propag. Lett. 3(1), 141–144 (2004)Google Scholar
  11. 11.
    Gonzáles García, S., Godoy Rubio, R., Rubio Bretones, A., Gómez Martín, R.: On the dispersion relation of ADI-FDTD. IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006)CrossRefGoogle Scholar
  12. 12.
    Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg (2006)Google Scholar
  13. 13.
    Hansen, E., Ostermann, A.: Dimension splitting for evolution equations. Numer. Math. 108, 557–570 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lee, J., Fornberg, B.: A split step approach for the 3-D Maxwell’s equations. J. Comput. Appl. Math. 158(2), 485–505 (2003)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lee, J., Fornberg, B.: Some unconditionally stable time stepping methods for the 3D Maxwell’s equations. J. Comput. Appl. Math. 166(2), 497–523 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Leis, R.: Initial-Boundary Value Problems in Mathematical Physics. B. G. Teubner, Stuttgart (1986)CrossRefMATHGoogle Scholar
  17. 17.
    Namiki, T.: A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory Tech. 47(10), 2003–2007 (1999)CrossRefGoogle Scholar
  18. 18.
    Namiki, T.: 3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations. IEEE Trans. Microw. Theory Tech. 48(10), 1743–1748 (Oct 2000)Google Scholar
  19. 19.
    Rauch, J.: Partial Differential Equations, Volume 128 of Graduate Texts in Mathematics. Springer, New York (1991)Google Scholar
  20. 20.
    Reed, M., Simon, B.: Methods of modern mathematical physics I. Functional Analysis, 2nd edn. Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York (1980)Google Scholar
  21. 21.
    Sz-Nagy, B.: Spektraldarstellung Linearer Transformationen des Hilbertschen Raumes. Berichtigter Nachdruck, Volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin, New York (1967) (reprint of the 1942 original edition edition)Google Scholar
  22. 22.
    Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edn. Artech House Publishers, Norwood (2005)Google Scholar
  23. 23.
    Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. Birkhäuser, Basel (2009)CrossRefMATHGoogle Scholar
  24. 24.
    Verwer, J.G.: Component splitting for semi-discrete Maxwell equations. BIT 51(2), 427–445 (2011)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Verwer, J.G.: Composition methods, Maxwell’s equations, and source terms. SIAM J. Numer. Anal. 50(2), 439–457 (2012)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Verwer, J.G., Botchev, M.A.: Unconditionally stable integration of Maxwell’s equations. Linear Algebra Appl. 431(3–4), 300–317 (2009)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966)CrossRefMATHGoogle Scholar
  28. 28.
    Zheng, F., Chen, Z., Zhang, J.: Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method. IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Marlis Hochbruck
    • 1
  • Tobias Jahnke
    • 1
  • Roland Schnaubelt
    • 1
  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations