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Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium

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Abstract

This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (PEC) boundary condition, employing the energy method. To this end, we first establish some new energy identities for the 2D Maxwell equations in a lossy medium with a PEC boundary condition. Then by making use of these energy identities, it is proved that the FDTD scheme and its time difference scheme are stable in the discrete L 2 and H 1 norms when the CFL condition is satisfied. It is shown further that the solution to both the FDTD scheme and its time difference scheme is second-order convergent in both space and time in the discrete L 2 and H 1 norms under a slightly stricter condition than the CFL condition. This means that the solution to the FDTD scheme is superconvergent. Numerical results are also provided to confirm the theoretical analysis.

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References

  1. Berenger J P. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys, 1994, 114: 185–200

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen W, Li X, Liang D. Energy-conserved splitting FDTD Schemes for Maxwell’s equations. Numer Math, 2008, 108: 445–485

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen W, Li X, Liang D. Energy-conserved splitting finite difference time-domain methods for Maxwell’s equations in three dimensions. SIAM J Numer Anal, 2010, 108: 445–485

    Google Scholar 

  4. Chew W C, Weedon W H. A 3-D perfectly matched medium for the modified Maxwell equations with streched coordinates. Micro Opt Tech Lett, 1994, 7: 257–260

    Article  Google Scholar 

  5. Chung E T, Du Q, Zou J. Convergence analysis of a finite volume method for Maxwell’s equations in nonhomogeneous media. SIAM J Numer Anal, 2003, 41: 37–63

    Article  MATH  MathSciNet  Google Scholar 

  6. Engqusit B, Majda A. Absorbing boundary conditions for the numerical simulation of waves. Math Comp, 1977, 31: 629–651

    Article  MathSciNet  Google Scholar 

  7. Fang N S, Ying L A. Stability analysis of FDTD to UPML for time dependent Maxwell equations. Sci China Ser A, 2009, 52: 794–1811

    Article  MATH  MathSciNet  Google Scholar 

  8. Gao L. Splitting finite difference time domain methods for time-domain Maxwell equations. PhD Thesis. Coventry, UK: Coventry University, 2006

    Google Scholar 

  9. Gao L, Zhang B, Liang D. The splitting finite difference time domain methods for Maxwell’s equations in two dimensions. J Comput Appl Math, 2007, 205: 207–230

    Article  MATH  MathSciNet  Google Scholar 

  10. Gao L, Zhang B, Liang D. Splitting finite difference methods on staggered grids for the three dimensional time dependent Maxwell equations. Commun Comput Phys, 2008, 4: 405–432

    MathSciNet  Google Scholar 

  11. Ge D B, Yan Y B. Finite Difference Time Domain Method for Electromagnetic Waves (in Chinese). Xi’an: XiDian Univ Press, 2005

    Google Scholar 

  12. Kong L, Hong J, Zhang J. Splitting multisymplectic integrators for Maxwell’s equations. J Comput Phys, 2010, 229: 4259–4278

    Article  MATH  MathSciNet  Google Scholar 

  13. Leis R. Initial Boundary Value Problems in Mathematical Physics. New York: John Wiley, 1986

    MATH  Google Scholar 

  14. Levy D, Tadmor E. From semidiscrete to fully discrete stability of Runge-Kutta schemes by the energy method. SIAM Rev, 1998, 40: 40–73

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin Q, Li J. Superconvergence analysis for Maxwell’s equations in dispersive media. Math Comput, 2008, 77: 757–771

    MATH  MathSciNet  Google Scholar 

  16. Lin Q, Yan N. Global superconvergence for Maxwell’s equations. Math Comp, 1999, 69: 159–176

    Article  MathSciNet  Google Scholar 

  17. Liu Y. Fourier analysis of numerical algorithms for the Maxwell equations. J Compu Phys, 1996, 124: 396–416

    Article  MATH  Google Scholar 

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

    Google Scholar 

  19. Min M S, Teng C H. The instability of the Yee scheme for the “magic time step”. J Comput Phys, 2001, 166: 418–424

    Article  MATH  MathSciNet  Google Scholar 

  20. Monk P. Finite Element Methods for Maxwell’s Equations. Oxford: Clarendon Press, 2003

    Book  MATH  Google Scholar 

  21. Monk P, Süli E. A convergence analysis of Yee’s scheme on nonuniform grid. SIAM J Numer Anal, 1994, 31: 393–412

    Article  MATH  MathSciNet  Google Scholar 

  22. Monk P, Süli E. Error estimates for Yee’s method on nonuniform grids. IEEE Trans Magnetics, 1994, 30: 3200–3203

    Article  Google Scholar 

  23. Mur G. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations. IEEE Trans Electr Comput, 1981, 23: 377–382

    Article  Google Scholar 

  24. Namiki T. A new FDTD algorithm based on alternating direction implicit method. IEEE Trans Microwave Theory Tech, 1999, 47: 2003–2007

    Article  Google Scholar 

  25. Nicolaides R A, Wang D Q. Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions. Math Comput, 1998, 67: 947–963

    Article  MATH  MathSciNet  Google Scholar 

  26. Petropoulos P G, Zhao L, Cangellaris A C. A reflectionless sponge layer absorbing boundary condition for the solution of Maxwell’s equations with high-order staggered finite difference schemes. J Comput Phys, 1998, 139: 184–208

    Article  MATH  MathSciNet  Google Scholar 

  27. Remis R F. On the stability of the finite-difference time-domain method. J Compu Phys, 2000, 163: 249–261

    Article  MATH  MathSciNet  Google Scholar 

  28. Remis R F. Stability of FDTD on nonuniform grids for Maxwell’s equations in lossless media. J Comput Phys, 2006, 218: 594–606

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  30. Taflove A, Brodwin M E. Numerical solution of steady-state electromagnetic scattering problems using the timedependent Maxwell equations. IEEE Trans Microwave Theory Tech, 1975, 23: 623–630

    Article  Google Scholar 

  31. Taflove A, Hagness S. Computational Electrodynamics: The Finite-Difference Time-Domain Method (2nd ed). Boston: Artech House, 2000

    MATH  Google Scholar 

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

    Google Scholar 

  33. Yee K S. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans Antennas Propagat, 1966, 14: 302–307

    Article  MATH  Google Scholar 

  34. Zheng F, Chen Z, Zhang J. Toward the development of a three-dimensional unconditionally stable finite difference time-domain method. IEEE Trans Microwave Theory Tech, 2000, 48: 1550–1558

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computational and Applied Mathematics, School of Sciences, China University of Petroleum, Qingdao, 266555, China

    LiPing Gao

  2. LSEC and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Bo Zhang

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  1. LiPing Gao
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  2. Bo Zhang
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Correspondence to LiPing Gao.

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Gao, L., Zhang, B. Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium. Sci. China Math. 54, 2693–2712 (2011). https://doi.org/10.1007/s11425-011-4305-7

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  • DOI: https://doi.org/10.1007/s11425-011-4305-7

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