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An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media

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Abstract

In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields, and the induced electric polarization (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then, we solve the IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show that such an enriched collocation method is convergent and accurate.

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References

  1. Baker, G.A.: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 564–576 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banks, H.T., Bokil, V.A., Gibson, N.L.: Analysis of stability and dispersion in a finite element method for debye and lorentz media. Numer. Methods Partial Differ. Equ. 25, 885–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001)

    MATH  Google Scholar 

  4. Brunner, H.: Collocation Methods for Volterra and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  5. Cannarsa, P., Sforza, D.: A stability result for a class of nonlinear integro-differential equations with l 1 kernels. Appl. Math. (Warsaw) 35, 395–430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Scientific Computation. Springer, Berlin (2006). Fundamentals in single domains

    MATH  Google Scholar 

  7. Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364–381 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Causley, M.: Asymptotic and numerical analysis of time-dependent wave propagation in dispersive dielectric media that exhibit fractional relaxation. Ph.D. dissertation, Rutgers-Newark (2011)

  9. Cole, K., Cole, R.: Dispersion and absorption in dielectrics, I: alternating current characteristics. J. Chem. Phys. 9(4), 341–352 (1941)

    Article  Google Scholar 

  10. Cooper, J., Marx, B., Buhl, J., Hombach, V.: Determination of safety distance limits for a human near a cellular base station antenna, adopting the IEEE standard or ICNIRP guidelines. Bioelectromagnetics 23, 429–443 (2000)

    Article  Google Scholar 

  11. Feng, K., Qin, M.Z.: Symplectic Geometric Algorithms for Hamiltonian Systems. Zhejiang Science & Technology Press, Hangzhou (2003)

    Google Scholar 

  12. Gabriel, C., Gabriel, S.: Compilation of the dielectric properties of body tissues at RF and microwave frequencies. U.S.A.F Armstrong Lab, Brooks AFB, TX, Technical Rep. AL/OE-TR-1996-0037 (1996)

  13. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  14. Huang, C., Stynes, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42(5), 1015–1030 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiao, D., Jin, J.M.: Time-domain finite element modeling of dispersive media. IEEE Microw. Wirel. Compon. Lett. 11, 220–222 (2001)

    Article  Google Scholar 

  16. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)

    Article  MathSciNet  Google Scholar 

  17. Kim, T., Kim, J., Pack, J.: Dispersive effect of UWB pulse on human head. Electromagnetic Compatibility. In: EMC Zurich, 18th International Zurich Symposium (2007)

  18. Lason, S., Saedpanah, F.: The continuous Galerkin method for an integro-differential equation modeling fractional dynamic order viscoelasticity. IMA J. Numer. Anal. 30, 964–986 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, J.: A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput. 31, 4696–4714 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, J.C., Huang, Y.Q., Lin, Y.P.: Developing finite element methods for Maxwell’s equations in a Cole-Cole dispersive medium. SIAM J. Sci. Comput. 33(6), 3153–3174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, W.B., Shen, J.: A new efficient spectral-Galerkin method for singular perturbation problems. J. Sci. Comput. 11, 411–437 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, W.B., Tang, T.: Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems. Appl. Numer. Math. 38(3), 315–345 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. McLean, W.: Fast summation by interval clustering for an evolution equation with memory. SIAM J. Sci. Comput. 34(6), A3039–A3056 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  26. Rekanos, I., Papadopoulos, T.: An auxiliary differential equation method for FDTD modeling of wave propagation in Cole-Cole dispersive media. IEEE Trans. Antennas Propag. 58(11), 3666–3674 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rekanos, I., Yioultsis, T.: Approximation of Grüwald-Letnikov fractional derivative for FDTD modeling of Cole-Cole media. IEEE Trans. Magn. 50(2), 7004304 (2014)

    Article  Google Scholar 

  28. Repo, T., Pulli, S.: Application of impedance spectroscopy for selecting frost hardy varieties of English ryegrass. Ann. Bot. 78, 605–609 (1996)

    Article  Google Scholar 

  29. Saedpanah, F.: Well-posedness of an integro-differential equation with positive type kernels modeling fracional order viscoelasiticity. Eur. J. Mech. A-Solid 44, 201–211 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shelby, R.A., Smith, D.R., Nemat-Nasser, S.C., Schultz, S.: Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial. Appl. Phys. Lett. 78, 489–491 (2001)

    Article  Google Scholar 

  31. Shen, J.: Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and applications, volume 41 of Series in Computational Mathematics. Springer, Berlin (2011)

    Book  Google Scholar 

  33. Shen, J., Wang, L.L.: Fourierization of the Legendre-Galerkin method and a new space-time spectral method. Appl. Numer. Math. 57, 710–720 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Smith, D.R., Kroll, N.: Negative refractive index in left-handed materials. Phys. Rev. Lett. 85, 2933–2936 (2000)

    Article  Google Scholar 

  35. Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton University Press, Princeton and Oxford (2002)

    MATH  Google Scholar 

  36. Strauss, W.: Partial Differential Equations: an Introduction, 2nd edn. Wiley, Brown University (2007)

  37. Tofighi, M.: FDTD modeling of biological tissues Cole-Cole dispersion for 0.5–30 GHz using relaxation time distribution samples-novel and improved implementations. IEEE Microw. Theory Tech. 57(10), 2588–2596 (2009)

    Article  Google Scholar 

  38. Torres, F., Vaudon, P., Jecko, B.: Application of fractional derivatives to the FDTD modeling of pulse propagation in a Cole-Cole dispersive medium. Microw. Opt. Technol. Lett. 13/5, 300–304 (1996)

    Article  Google Scholar 

  39. Wang, B., Xie, Z., Zhang, Z.: Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media. J. Comput. Phys. 229, 8552–8563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, L.L., Samson, M.D., Zhao, X.D.: A well-conditioned collocation method using a pseudospectral integration matrix. SIAM J. Sci. Comput. 36(3), A907–A929 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, L.L., Shen, J.: Error analysis for mapped Jacobi spectral methods. J. Sci. Comput. 24, 183–218 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xiang, S.: On interpolation approximation: convergence rates for polynomial interpolation for functions of limited regularity. SIAM J. Numer. Anal. 54(4), 2081–2113 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yosida, K.: Functional Analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften, 6th edn. Springer, Berlin-New York (1980)

    Google Scholar 

  44. Ziolkowski, R.W.: Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs. Opt. Express 11, 662–681 (2003)

    Article  Google Scholar 

  45. Ziolkowski, R.W., Heyman, E.: Wave propagation in media having negative permittivity and permeability. Phys. Rev. E 64, 056625 (2003)

    Article  Google Scholar 

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Acknowledgments

The authors would like to thank both universities for hosting their mutual visits to complete this work.

Funding

The research of author Can Huang is supported by the National Natural Science Foundation of China (no. 11401500, 91630204, 11771363).

The research of author Li-Lian Wang is partially supported by Singapore MOE AcRF Tier 1 Grant (RG 15/12) and Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059).

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Fujian, 361005, China

    Can Huang

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Li-Lian Wang

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  1. Can Huang
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  2. Li-Lian Wang
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Correspondence to Li-Lian Wang.

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Communicated by: Jan Hesthaven

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Huang, C., Wang, LL. An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Adv Comput Math 45, 707–734 (2019). https://doi.org/10.1007/s10444-018-9636-2

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  • DOI: https://doi.org/10.1007/s10444-018-9636-2

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