Advertisement

Journal of Scientific Computing

, Volume 17, Issue 1–4, pp 405–422 | Cite as

Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics

  • S. Abarbanel
  • D. Gottlieb
  • J. S. Hesthaven
Article

Abstract

We investigate the long time behavior of two unsplit PML methods for the absorption of electromagnetic waves. Computations indicate that both methods suffer from a temporal instability after the fields reach a quiescent state. The analysis reveals that the source of the instability is the undifferentiated terms of the PML equations and that it is associated with a degeneracy of the quiescent systems of equations. This highlights why the instability occurs in special cases only and suggests a remedy to stabilize the PML by removing the degeneracy. Computational results confirm the stability of the modified equations and is used to address the efficacy of the modified schemes for absorbing waves.

computational electromagnetics PML stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Engquist, B., and Majda, A. (1977). Absorbing boundary conditions for the numerical solution of waves. Math. Comp. 31, 629–651.Google Scholar
  2. 2.
    Bayliss, A., and Turkel, E. (1980). Radiation boundary conditions for wave-like equations. Comm. Pure Appl. Math. 23, 707–725.Google Scholar
  3. 3.
    Givoli, D. (1991). Nonreflecting boundary conditions. J. Comput. Phys. 94, 1–29.Google Scholar
  4. 4.
    Grote, M., and Keller, J. (1996). Nonreflecting boundary conditions for time dependent scattering. J. Comput. Phys. 127, 52–81.Google Scholar
  5. 5.
    Tsynkov, S. V. (1998). Numerical solution of problems on unbounded domains. Appl. Numer. Math. 27, 465–532.Google Scholar
  6. 6.
    Hagstrom, T. (1999). Radiation boundary conditions for the numerical simulation of waves. Acta Numerica 8, 47–106.Google Scholar
  7. 7.
    Berenger, J. P. (1994). A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200.Google Scholar
  8. 8.
    Berenger, J. P. (1996). Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127, 363–379.Google Scholar
  9. 9.
    Hu, F. Q. (1996). On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys. 129, 201–219.Google Scholar
  10. 10.
    Chew, W. C., and Liu, Q. H. (1996). Perfectly matched layers for elastodynamics: A new absorbing boundary condition. J. Comput. Acoust. 4, 341–349.Google Scholar
  11. 11.
    Abarbanel, S., and Gottlieb, D. (1997). A mathematical analysis of the PML method. J. Comput. Phys. 134, 357–363.Google Scholar
  12. 12.
    Hesthaven, J. S. (1998). On the analysis and construction of perfectly matched layers for the linearized Euler equations. J. Comput. Phys. 142, 129–147.Google Scholar
  13. 13.
    Gedney, S. D. (1996). Anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas Propagat. 44, 1630–1639.Google Scholar
  14. 14.
    Ziolkowski, R. W. (1997). Time-derivative Lorenz material model based absorbing boundary conditions. IEEE Trans. Antennas Propagat. 45, 1530–1535.Google Scholar
  15. 15.
    Abarbanel, S., and Gottlieb, D. (1998). On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math. 27, 331.Google Scholar
  16. 16.
    Hardin, J. C., Ristorcelli, J. R., and Tam, C. K. W. (eds.) (1995). ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA), NASA CP 3300, NASA Langley Research Center, Virginia.Google Scholar
  17. 17.
    Abarbanel, S., Gottlieb, D., and Hesthaven, J. S. (1999). Well-posed perfectly matched layers for advective acoustics. J. Comput. Phys. 154, 266–283.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • S. Abarbanel
    • 1
  • D. Gottlieb
    • 2
  • J. S. Hesthaven
    • 2
  1. 1.Department of Applied MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.Division of Applied MathematicsBrown UniversityProvidence

Personalised recommendations