Advertisement

Some Numerical Techniques for Maxwell’s Equations in Different Types of Geometries

  • Bengt Fomberg
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 31)

Summary

Almost all the difficulties that arise in finite difference time domain solutions of Maxwell's equations are due to material interfaces (to which we include objects such as antennas, wires, etc.) Different types of difficulties arise if the geometrical features are much larger than or much smaller than a typical wave length. In the former case, the main difficulty has to do with the spatial discretisation, which needs to combine good geometrical flexibility with a relatively high order of accuracy. After discussing some options for this situation, we focus on the tatter case. The main problem here is to find a time stepping method which combines a very low cost per time step with unconditional stability. The first such method was introduced in 1999 and is based on the ADI principle. We will here discuss that method and some subsequent developments in this area.

Key words

Maxwell’s equations FDTD ADI split step pseudospectral methods finite differences spectral elements 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Bruno, New high-order, high-frequency methods in computational electromagnetism. Topics in Computational Wave Propagation 2002, Springer (2003).Google Scholar
  2. 2.
    G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53.MathSciNetzbMATHGoogle Scholar
  3. 3.
    G. Dahlquist, 33 years of numerical instability, part I, BIT, 25 (1985), 188–204.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Darms, R. Schuhmann, H. Spachmann and T. Weiland, Asymmetry effects in the ADI-FDTD algorithm, to appear in IEEE Microwave Guided Wave Lett.Google Scholar
  5. 5.
    L. Demkowicz, Fully automatic hp-adaptive finite elements for the time-harmonic Maxwell’s equations. Topics in Computational Wave Propagation 2002, Springer (2003).Google Scholar
  6. 6.
    J. Douglas, Jr, On the numerical integration of \( \frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 u}} {{\partial y^2 }} = \frac{{\partial u}} {{\partial t}} \) by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42–65.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T.A. Driscoll and B. Fomberg, Block pseudospectral methods for Maxwell’s equations: II. Two—dimensional, discontinuous—coefficient case, SIAM Sci. Comput. 21 (1999), 1146–1167.CrossRefzbMATHGoogle Scholar
  8. 8.
    F. Edelvik and G. Ledfelt, A comparison of time-domain hybrid solvers for complex scattering problems. Int. J. Numer. Model 15 (5–6) (2002), 475–487.CrossRefzbMATHGoogle Scholar
  9. 9.
    M. El Hachemi, Hybrid methods for solving electromagnetic scattering problems on over-lapping grids, in preparation.Google Scholar
  10. 10.
    E. Forest and R.D. Ruth, Fourth order symplectic integration, Physica D 43 (1990), 105–117.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press (1996).Google Scholar
  12. 12.
    B. Fornberg, Calculation of weights in unite difference formulas, SIAM Review, 40 (1998), 685–691.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Fornberg, A short proof of the unconditional stability of the ADI-FDTD scheme. Uni versity of Colorado, Department of Applied Mathematics Technical Report 472 (2001).Google Scholar
  14. 14.
    B. Fornberg, High order finite differences and the pseudospectral method on staggered grids, SIAM J. Numer. Anal. 27 (1990), 904–918.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    B. Fornberg and T.A. Driscoll, A fast spectral algorithm for nonlinear wave equations with linear dispersion, JCP, 155 (1999), 456–467.MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. Fomberg and M. Christ, Spatial finite difference approximations for wave-type equations, SIAM J. Numer. Anal. 37 (1999), 105–130.MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Frank and C.W. Ueberhuber, Iterated defect correction for diferential-equations 1. Theoretical results, Computing, 20 Nr 3 (1978), 207–228.MathSciNetzbMATHGoogle Scholar
  18. 18.
    R. Frank, F. Macsek and C.W. Ueberhuber, Iterated defect correction for diferential-equations 2. Numerical experiments, Computing, 33 Nr 2 (1984), 107–129.MathSciNetzbMATHGoogle Scholar
  19. 19.
    R. Frank, J. Hertling and H. Lehner, Defect correction algorithms for stiff ordinary differential equations. Computing, Supplement 5 (1984), 33–41.MathSciNetGoogle Scholar
  20. 20.
    L. Gao, B. Zhang and D. Liang, Stability and convergence analysis of the ADI-FDTD algorithm for 3D Maxwell equation. In preparation.Google Scholar
  21. 21.
    S.G. García, T.-W. Lee and S.C. Hagness, On the accuracy of the ADI-FDTD method, IEEE Antennas and Wireless Propagation Letters 1 No 1 (2002), 31–34.CrossRefGoogle Scholar
  22. 22.
    M. Ghrist, T.A. Driscoll and B. Fornberg, Staggered time integrators for wave equations, SIAM J. Num. Anal. 38 (2000), 718–741.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    B. Gustafsson and W. Kress, Deferred correction methods for initial value problems, BIT 41 (2001), 986–995.MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Verlag (2002).Google Scholar
  25. 25.
    J. Hesthaven and T. Warburton, Nodal high—order methods on unstructured grids I. Time-domain solution of Maxwell’s equations, JCP, 181 (2002), 186–221.MathSciNetzbMATHGoogle Scholar
  26. 26.
    R. Hiptmair, Finite elements in computational electromagnetism. Acta Numerica 2002 (2002), 237–339.Google Scholar
  27. 27.
    J. Jin, The Finite Element Method in Electromagnetics, Wiley, New York (1993).zbMATHGoogle Scholar
  28. 28.
    K.S. Kunz and J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press, Inc. (1993).Google Scholar
  29. 29.
    J. Lee and B. Fornberg, A split step approach for the 3D Maxwell’s equations. University of Colorado, Department of Applied Mathematics Technical Report 471 (2001), submitted to Journal of Computational and Applied Mathematics (2002).Google Scholar
  30. 30.
    J, Lee and B. Fornberg, Some unconditionally stable time stepping methods for the 3D Maxwell’s equations. Submitted to IMA journal of Applied Mathematics (2002).Google Scholar
  31. 31.
    G. Liu and S.D. Gedney, Perfectly matched layer media for an unconditionally stable three-dimensional ADI-FDTD method, IEEE Microwave Guided Wave Lett. 10 (2000), 261–263.CrossRefGoogle Scholar
  32. 32.
    J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press, Oxford (1873).Google Scholar
  33. 33.
    T. Namiki, 3D ADI-FDTD method-Unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations, IEEE Transactions on Microwave Theory and Techniques, 48, No 10 (2000), 1743–1748.CrossRefGoogle Scholar
  34. 34.
    F. Neri, Lie algebras and canonical integration. Dept. of Physics, University of Maryland, preprint (1987).Google Scholar
  35. 35.
    D. Peaceman and J.H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    V. Pereyra, Accelerating the convergence of discretization algorithms, SIAM J. Numer. Anal. 4 (1967), 508–532.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    L.F. Richardson, The deferred approach to the limit, Phil. Trans. A, 226 (1927), 299–349.CrossRefzbMATHGoogle Scholar
  38. 38.
    T. Rylander and A. Bondeson, Stable FEM-FDTD hybrid method for Maxwell’s equations, Computer Physics Communications, 125 (2000), 75–82.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    B. Shanker, A.A. Ergin, K. Aygun. E Michielssen, Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time—domain algorithm, IEEE Trans. Antennas Propagation 48 (2000), 510–523.MathSciNetCrossRefGoogle Scholar
  40. 40.
    G. Strang, On construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–516.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys. 32 (1991), 400–407.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time—Domain Method, 2nd ed., Artech House, Norwood (2000).zbMATHGoogle Scholar
  43. 43.
    V.S. Varadarajan, Lie groups, Lie algebras and their representation. Prentice Hall, Englewood Cliffs (1974).Google Scholar
  44. 44.
    K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagation, 14 (1966), 302–307.zbMATHGoogle Scholar
  45. 45.
    H. Yoshida, Construction of higher order symplectic integrators. Physics Letters A, 150 (1990), 262–268.MathSciNetCrossRefGoogle Scholar
  46. 46.
    F. Zheng, Z. Chen, J. Zhang, A finite—difference time—domain method without the Courant stability conditions, IEEE Microwave Guided Wave Lett. 9 (1999), 441–443.CrossRefGoogle Scholar
  47. 47.
    F. Zheng, Z. Chen, J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE Transactions on Microwave Theory and Techniques, 48, No 9 (2000), 1550–1558.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Bengt Fomberg
    • 1
  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

Personalised recommendations