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Energy Decay and Stability of a Perfectly Matched Layer For the Wave Equation

  • Daniel H. BaffetEmail author
  • Marcus J. Grote
  • Sébastien Imperiale
  • Maryna Kachanovska
Article
  • 23 Downloads

Abstract

In Grote and Sim (Efficient PML for the wave equation. Preprint, arXiv:1001.0319 [math:NA], 2010; in: Proceedings of the ninth international conference on numerical aspects of wave propagation (WAVES 2009, held in Pau, France, 2009), pp 370–371), a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and \(L^2\) stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations with constant damping coefficients. Numerical results validate the theory.

Keywords

Perfectly matched layers Stability Numerical stability 

Notes

Acknowledgements

The fourth author acknowledges the partial support of a public Grant as part of the Investissement d’avenir Project, Reference ANR-11-LABX-0056-LMH, LabEx LMH.

References

  1. 1.
    Abarbanel, S., Gottlieb, D.: A mathematical analysis of the PML method. J. Comput. Phys. 134, 357–363 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abarbanel, S., Gottlieb, D., Hestahaven, J.S.: Well-posed perfectly matched layers for advective acoustics. J. Comput. Phys. 154, 266–283 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Appelö, D., Hagstrom, T., Kreiss, G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J. Appl. Math. 67(1), 1–23 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Appelö, D., Kreiss, G.: Application of a perfectly matched layer to the nonlinear wave equation. Wave Motion 44, 531–548 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barucq, H., Diaz, J., Tlemcani, M.: New absorbing layers conditions for short water waves. J. Comput. Phys. 229(1), 58–72 (2010).  https://doi.org/10.1016/j.jcp.2009.08.033 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bécache, E., Fauqueux, S., Joly, P.: Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 399–433 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bécache, E., Joly, P.: On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations. M2AN Math. Model. Numer. Anal. 36(1), 87–119 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bécache, E., Joly, P., Kachanovska, M.: Stable perfectly matched layers for a cold plasma in a strong background magnetic field. J. Comput. Phys. 341, 76–101 (2017).  https://doi.org/10.1016/j.jcp.2017.03.051 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bécache, É., Joly, P., Kachanovska, M., Vinoles, V.: Perfectly matched layers in negative index metamaterials and plasmas. CANUM 2014–42e Congrès National d’Analyse Numérique, ESAIM Proc. Surveys, vol. 50, pp. 113–132. EDP Sci, Les Ulis (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bécache, E., Joly, P., Vinoles, V.: On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials. Math. Comput. 87(314), 2775–2810 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bécache, E., Kachanovska, M.: Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: necessary and sufficient conditions of stability. ESAIM Math. Model. Numer. Anal. 51(6), 2399–2434 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bécache, E., Petropoulos, P.G., Gedney, S.D.: On the long-time behavior of unsplit perfectly matched layers. IEEE Trans. Antennas Propag. 52(5), 1335–1342 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bérenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bérenger, J.P.: Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127(2), 363–379 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chabassier, J., Imperiale, S.: Space/time convergence analysis of a class of conservative schemes for linear wave equations. C. R. Math. Acad. Sci. Paris 355(3), 282–289 (2017).  https://doi.org/10.1016/j.crma.2016.12.009 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, Z.: Convergence of the time-domain perfectly matched layer method for acoustic scattering problems. Int. J. Numer. Anal. Model. 6(1), 124–146 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chew, W.C., Weedon, W.H.: A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw. Opt. Technol. Lett. 7(13), 599–604 (1994)CrossRefGoogle Scholar
  18. 18.
    Collino, F., Monk, P.B.: Optimizing the perfectly matched layer. Comput. Methods Appl. Mech. Eng. 164(1–2), 157–171 (1998).  https://doi.org/10.1016/S0045-7825(98)00052-8. Exterior problems of wave propagation (Boulder, CO, 1997; San Francisco, CA, 1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Collino, F., Tsogka, C.: Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66, 294–307 (2001)CrossRefGoogle Scholar
  20. 20.
    Demaldent, E., Imperiale, S.: Perfectly matched transmission problem with absorbing layers: application to anisotropic acoustics in convex polygonal domains. Int. J. Numer. Methods Eng. 96(11), 689–711 (2013).  https://doi.org/10.1002/nme.4572 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Diaz, J., Joly, P.: A time domain analysis of PML models in acoustics. Comput. Methods Appl. Mech. Eng. 195(29–32), 3820–3853 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Duru, K.: The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38(2), A1171–A1194 (2016).  https://doi.org/10.1137/140976443 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Duru, K., Kreiss, G.: Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form. SIAM Numer. Anal. 52, 2883–2904 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ervedoza, S., Zuazua, E.: Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves. Numer. Math. 109(4), 597–634 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Grote, M., Sim, I.: Efficient PML for the wave equation. Preprint. arXiv:1001.0319 [math:NA] (2010)
  26. 26.
    Grote, M.J., Sim, I.: Perfectly matched layer for the second-order wave equation. In: Proceedings of the Ninth International Conference on Numerical Aspects of Wave Propagation (WAVES 2009, held in Pau, France, 2009), pp. 370–371Google Scholar
  27. 27.
    Hagstrom, T., Appelö, D.: Automatic symmetrization and energy estimates using local operators for partial differential equations. Commun. Partial Differ. Equ. 32(7–9), 1129–1145 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hu, F.Q.: On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys. 129, 201–219 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hu, F.Q.: A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables. J. Comput. Phys. 173(2), 455–480 (2001).  https://doi.org/10.1006/jcph.2001.6887 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Joly, P.: An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation. SeMA J. 57, 5–48 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kachanovska, M.: Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: energy estimates (2017). https://hal.inria.fr/hal-01419682. Https://hal.inria.fr/hal-01419682
  32. 32.
    Kaltenbacher, B., Kaltenbacher, M., Sim, I.: A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics. J. Comput. Phys. 235, 407–422 (2013).  https://doi.org/10.1016/j.jcp.2012.10.016 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Komatitsch, D., Tromp, J.: A perfectly matched layer absorbing boundary conditionfor the second-order seismic wave equation. Geophys. J. Int. 154, 146–153 (2003)CrossRefGoogle Scholar
  34. 34.
    Nataf, F.: A new approach to perfectly matched layers for the linearized Euler system. J. Comput. Phys. 214(2), 757–772 (2006).  https://doi.org/10.1016/j.jcp.2005.10.014 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sjögreen, B., Petersson, N.A.: Perfectly matched layers for Maxwell’s equations in second order formulation. J. Comput. Phys. 209, 19–46 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhao, L., Cangellaris, A.C.: Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids. IEEE Trans. Microw. Theory Tech. 44(12), 2555–2563 (1996)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Daniel H. Baffet
    • 1
    Email author
  • Marcus J. Grote
    • 1
  • Sébastien Imperiale
    • 2
  • Maryna Kachanovska
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland
  2. 2.Inria — LMS, Ecole Polytechnique, CNRS — Institut Polytechnique de ParisPalaiseauFrance
  3. 3.POEMS (UMR 7231 CNRS, ENSTA, INRIA), INRIA, Institut Polytechnique de ParisPalaiseauFrance

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