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Stability analysis of FDTD to UPML for time dependent Maxwell equations

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Abstract

We study an finite-difference time-domain (FDTD) system of uniaxial perfectly matched layer (UPML) method for electromagnetic scattering problems. Particularly we analyze the discrete initial-boundary value problems of the transverse magnetic mode (TM) to Maxwell’s equations with Yee’s algorithm. An exterior domain in two spacial dimension is truncated by a square with a perfectly matched layer filled by a certain artificial medium. Besides, an artificial boundary condition is imposed on the outer boundary of the UPML. Using energy method, we obtain the stability of this FDTD system on the truncated domain. Numerical experiments are designed to approve the theoretical analysis.

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References

  1. Taflov A. Computational Electrodynamics, The Finite Dependent Time Domain Approach (2nd ed). Norwood: Artech House, 2000

    Google Scholar 

  2. Lu Y H. Numerical Methods for Computational Electromagnetics (in Chinese). Beijing: Tsinghua University Press, 2006

    Google Scholar 

  3. Wang C Q. Computational Advanced Electromagnetics (in Chinese). Beijing: Peking University Press, 2005

    Google Scholar 

  4. Wang B Z. Computational Electromagnetics (in Chinese). Beijing: Science Press, 2002

    Google Scholar 

  5. Sheng X Q. Abstract of Computational Electromagnetics (in Chinese). Beijing: Science Press, 2004

    Google Scholar 

  6. Yee K S. Numerical solution of initial boundary value problems involving Maxwell’s equations. IEEE Trans Antennas Propag, 14: 302–307 (1966)

    Article  MATH  Google Scholar 

  7. Ying L. Numerical Methods for Exterior Problems, Singapore: World Scientific, 2006

    MATH  Google Scholar 

  8. Bérenger J P. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys, 114: 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bérenger J P. Perfectly matched layer for the FDTD solution of wave-structure interaction problems. IEEE Trans Anttennas Propag, 44: 110–117 (1996)

    Article  Google Scholar 

  10. Bérenger J P. Three dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys, 127: 363–379 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bérenger J P. Numerical reflection from FDTD-PMLs: a comparison of the split PML with the unsplit and CFS PMLs. IEEE Trans Anttennas Propag, 50(3): 258–265 (2002)

    Article  Google Scholar 

  12. Abarbanel S A, Gottlieb D. A mathematical analysis of the PML method. J Comput Phys, 134: 357–363 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chew W, Weedon W. A 3rd perfectly matched medium for modified Maxwell’s equations with streched coordinates. Microwave Opt Techno Lett, 13(7): 599–604 (1994)

    Article  Google Scholar 

  14. Ying L. Analysis of the PML problems for time dependent Maxwell equations. Far East J Appl Math, 31(3): 299–320 (2008)

    MathSciNet  Google Scholar 

  15. Becache E, Joly P. On the analysis of Berenger’s PML for Maxwell’s equations. Math Model Anal, 36(1): 87–120 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fang N S, Ying L. Stability analysis of Yee schemes to PML and UPML for Maxwell equations. J Comput Math, to appear, 2009

  17. Bramble J H, Pasciak J E. Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math Comp, 76: 597–614 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bao G, Wu H. Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations. SIAM J Numer Anal, 43(5): 2121–2143 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen Z, Liu X. An adaptive perfectly matched layers technique for time-harmonic scattering problems. SIAM J Numer Anal, 43: 645–671 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen Z, Wu H. An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J Numer Anal, 41: 799–826 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lu T, Zhang P, Cai W. Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions. J Comput Phys, 200(2): 549–580 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kung F, Chuan H T. Stability of classicla FDTD formulation with nonlinear elements: a new perspective. J Electromagn Waves Appl, 17(9): 1313–1314 (2003)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Finance, Southwestern University of Finance and Economics, Chengdu, 610074, China

    NengSheng Fang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    NengSheng Fang

  3. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Lung-An Ying

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  1. NengSheng Fang
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  2. Lung-An Ying
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Correspondence to Lung-An Ying.

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Fang, N., Ying, LA. Stability analysis of FDTD to UPML for time dependent Maxwell equations. Sci. China Ser. A-Math. 52, 794–816 (2009). https://doi.org/10.1007/s11425-009-0015-9

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  • DOI: https://doi.org/10.1007/s11425-009-0015-9

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