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A second-order space-time accurate scheme for Maxwell’s equations in a Cole–Cole dispersive medium

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Abstract

A fully implicit finite-difference time-domain (FDTD) scheme with second-order space-time accuracy for Maxwell’s equations in a Cole–Cole dispersive medium is proposed and analyzed. The temporal discretization is built upon Crank-Nicolson method, and the Caputo derivative term is based on the recently established \(\mathcal {L}2\text {-}1_{\sigma }\) formula and a weighted approach. A rigorous analysis is carried out to show that the proposed scheme is unconditionally stable and has second-order accuracy in both time and space. 2D and 3D numerical examples are presented to validate our theoretical findings.

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References

  1. Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics I. Alternating current characteristics. J Chem Phys 9:341–351

    Article  Google Scholar 

  2. Li J, Huang Y (2013) Time-domain finite element methods for Maxwell’s equations in metamaterials. Springer, Berlin

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Fan E, Wang J, Liu Y, Li H, Fang Z (2020) Numerical simulations based on shifted second-order difference/finite element algorithms for the time fractional Maxwell’s system. Eng Comput 1–15

  5. Huang C, Wang L (2019) An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Adv Comput Math 45:707–734

    Article  MathSciNet  MATH  Google Scholar 

  6. Gibson NL (2015) A polynomial Chaos method for dispersive electromagnetics. Commun Comput Phys 18:1234–1263

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang J, Zhang J, Zhang Z (2021) A CG-DG method for Maxwell’s equations in Cole-Cole dispersive media. J Comput Appl Math 393:113480

    Article  MathSciNet  MATH  Google Scholar 

  8. Yee K (1966) Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media. IEEE Trans Antennas Propag 14:302–307

    Article  MATH  Google Scholar 

  9. Tan EL (2008) Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods. IEEE Trans Antennas Propag 56:170–177

    Article  MathSciNet  MATH  Google Scholar 

  10. Jia H, Li J, Fang Z, Li M (2019) A new FDTD scheme for Maxwell’s equations in Kerr-type nonlinear media. Numer Algorithms 82:223–243

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang W, Liu L, Huang Y (2019) The FDTD simulation for the performance of dispersive cloak devices. Appl Math Lett 88:171–178

    Article  MathSciNet  MATH  Google Scholar 

  12. Li J, Shields S (2016) Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes. Numerische Mathematik 134:741–781

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang X, Li J, Fang Z (2018) Development and analysis of Crank-Nicolson scheme for metamaterial Maxwell’s equations on nonuniform rectangular grids. Numer Methods Partial Differ Equ 34:2040–2059

    Article  MathSciNet  MATH  Google Scholar 

  14. Bai X, Rui H (2021) New energy analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes. Adv Appl Math Mech 13:1355–1383

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen W, Li X, Liang D (2008) Energy-conserved splitting FDTD methods for Maxwell’s equations. Numerische Mathematik 108:445–485

    Article  MathSciNet  MATH  Google Scholar 

  16. Li W, Liang D (2020) The spatial fourth-order compact splitting FDTD scheme with modified energy-conserved identity for two-dimensional Lorentz model. J Comput Appl Math 367:112428

    Article  MathSciNet  MATH  Google Scholar 

  17. Gao L, Cao M, Shi R, Guo H (2019) Energy conservation and super convergence analysis of the EC-S-FDTD schemes for Maxwell equations with periodic boundaries. Numer Methods Partial Differ Equ 35:1562–1587

    Article  MathSciNet  MATH  Google Scholar 

  18. Liang D, Yuan Q (2013) The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations. J Comput Phys 243:344–364

    Article  MathSciNet  MATH  Google Scholar 

  19. Kong L, Hong Y, Tian N, Zhou W (2019) Stable and efficient numerical schemes for two-dimensional Maxwell equations in lossy medium. J Comput Phys 397:108703

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang Y, Chen M, Li J (2020) Development and analysis of both finite element and fourth-order in space finite difference methods for an equivalent Berengers PML model. J Comput Phys 405:109154

    Article  MathSciNet  MATH  Google Scholar 

  21. Gao L, Zhang B (2013) Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations. Sci China Math 56:1705–1726

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang Y, Chen M, Li J, Lin Y (2018) Numerical analysis of a leapfrog ADI-FDTD method for Maxwell’s equations in lossy media. Comput Math Appl 76:938–956

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang Y, Sun Z, Wu H (2011) Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation. SIAM J Numer Anal 49:2302–2322

    Article  MathSciNet  MATH  Google Scholar 

  24. Li Q, Chen Y, Huang Y, Wang Y (2020) Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method. Appl Numer Math 157:38–54

    Article  MathSciNet  MATH  Google Scholar 

  25. Fu H, Wang H (2019) A preconditioned fast parareal finite difference method for space-time fractional partial differential equation. J Sci Comput 78:1724–1743

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang Z, Zheng X, Wang H (2020) A variably distributed-order time-fractional diffusion equation: analysis and approximation. Comput Methods Appl Mech Eng 367:113118

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu Z, Li X, Huang J (2021) Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn equations. Numer Methods Partial Differ Equ 37:2613–2633

    Article  MathSciNet  Google Scholar 

  28. Guo B, Li J, Zmuda H (2006) A new FDTD formulation for wave propagation in biological media with Cole-Cole model. IEEE Microw Wirel Compon Lett 16:633–635

    Article  Google Scholar 

  29. Bia P, Caratelli D, Mescia L, Cicchetti R, Maione G, Prudenzano F (2015) A novel FDTD formulation based on fractional derivatives for dispersive Havriliak-Negami media. Signal Process 107:312–318

    Article  Google Scholar 

  30. Chakarothai J (2018) Novel FDTD scheme for analysis of frequency-dependent medium using fast inverse Laplace transform and Prony’s method. IEEE Trans Antennas Propag 67:6076–6089

    Article  Google Scholar 

  31. Bai X, Rui H (2021) An efficient FDTD algorithm for 2D/3D time fractional Maxwell’s system. Appl Math Lett 116:106992

    Article  MathSciNet  MATH  Google Scholar 

  32. Bai X, Wang S, Rui H (2021) Numerical analysis of Finite-Difference Time-Domain method for 2D/3D Maxwell’s equations in a Cole-Cole dispersive medium. Comput Math Appl 93:230–252

    Article  MathSciNet  MATH  Google Scholar 

  33. Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424–438

    Article  MathSciNet  MATH  Google Scholar 

  34. Ramezani M, Mokhtari R, Haase G (2020) Some high order formulae for approximating Caputo fractional derivatives. Appl Numer Math 153:300–318

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu Z, Li X, Zhang X (2020) A fast high-order compact difference method for the fractal mobile/immobile transport equation. Int J Comput Math 97:1860–1883

    Article  MathSciNet  MATH  Google Scholar 

  36. Yin B, Liu Y, Li H (2020) A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl Math Comput 368:124799

    MathSciNet  MATH  Google Scholar 

  37. Yin B, Liu Y, Li H, Zhang Z (2020) Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions. J Sci Comput 84:1–22

    Article  MathSciNet  MATH  Google Scholar 

  38. Li H, Cao J, Li C (2016) High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J Comput Appl Math 299:159–175

    Article  MathSciNet  MATH  Google Scholar 

  39. Petropoulos PG (2005) On the time-domain response of Cole-Cole dielectrics. IEEE Trans Antennas Propag 53:3741–3746

    Article  Google Scholar 

  40. Yan Y, Sun Z, Zhang J (2017) Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun Comput Phys 22:1028–1048

    Article  MathSciNet  MATH  Google Scholar 

  41. Lin Y, Xu C (2007) Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225:1533–1552

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and reviewers for their constructive comments and suggestions, which helped the authors to improve the quality of the paper significantly. The authors are very grateful to Dr. Jian Huang of Xiangtan University for his assistance in improve the quality of the paper. This work is supported by the National Natural Science Foundation of China Grant No. 12131014.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Xixian Bai

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Hongxing Rui

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  1. Xixian Bai
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  2. Hongxing Rui
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Contributions

XB: Conceptualization, Formal analysis, Methodology, Software, Writing-original draft. HR: Conceptualization, Funding acquisition, Investigation, Methodology, Supervision, Roles/Writing.

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Correspondence to Hongxing Rui.

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Bai, X., Rui, H. A second-order space-time accurate scheme for Maxwell’s equations in a Cole–Cole dispersive medium. Engineering with Computers 38, 5153–5172 (2022). https://doi.org/10.1007/s00366-021-01585-3

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  • DOI: https://doi.org/10.1007/s00366-021-01585-3

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