Skip to main content
SpringerLink
Log in

Highly Accurate Methods for Solving One-Dimensional Maxwell Equations in Stratified Media

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Earlier, a bicompact difference scheme was constructed for stationary and nonstationary Maxwell equations. Its stencil includes only one step of the spatial grid. A grid node is placed at each interface, and the other nodes may be placed arbitrarily. This scheme explicitly takes into account interface conditions on the interfaces. This makes it possible to compute generalized solutions with discontinuities of the solution and its derivatives. A novel spectral decomposition method is used for solving nonstationary problems that can take into account an arbitrary medium dispersion law. A new form of the bicompact scheme is proposed, which allows one to reduce the complexity of computations by a factor of four, which is a significant improvement. For the first time, a rigorous substantiation of the proposed scheme is given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. A. Belov and Zh. O. Dombrovskaya, “Bicompact finite-difference scheme for Maxwell’s equations in layered media,” Dokl. Math. 101, 185–188 (2020).

    Article  MATH  Google Scholar 

  2. W. N. Hansen, “Electric fields produced by the propagation of plane coherent electromagnetic radiation in a stratified medium,” J. Opt. Soc. Am. 58, 380–390 (1968).

    Article  Google Scholar 

  3. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation. J. Opt. Soc. Am. 62, 502–510 (1972).

    Article  Google Scholar 

  4. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A. 12, 1097–1103 (1995).

    Article  Google Scholar 

  5. A. V. Tishchenko, M. Hamdoun, and O. Parriaux, “Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam,” Opt. Quant. Electron. 35, 475–491 (2003).

    Article  Google Scholar 

  6. B. Fornberg, A Practical Guide to Pseudospectral Methods (University Press, Cambridge, 1996).

    Book  MATH  Google Scholar 

  7. T. Wriedt, Generalized Multipole Techniques for Electromagnetic and Light Scattering (Elsevier Science, Amsterdam, 1999).

    Google Scholar 

  8. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. ii. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A. 23, 2592–2601 (2006).

    Article  Google Scholar 

  9. M. A. Yurkin and M. Huntermann, “Rigorous and fast discrete dipole approximation for particles near a plane interface,” J. Phys. Chem. C. 119, 29088–29094 (2015).

    Article  Google Scholar 

  10. A. A. Shcherbakov and A. V. Tishchenko, “Generalized source method in curvilinear coordinates for 2d grating diffraction simulation, JQSRT 187, 76–96 (2017).

    Article  Google Scholar 

  11. M. Albani and P. Bernardi, “A numerical method based on the discretization of Maxwell equations in integral form,” IEEE Trans. Microwave Theory Techn. 22, 446–450 (1974).

    Article  Google Scholar 

  12. A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Techn. 35, 688–696 (1987).

    Article  Google Scholar 

  13. K. Beilenhoff, W. Heinrich, and H. L. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Techn. 40, 540–546 (1992).

    Article  Google Scholar 

  14. J.-S. Wang and R. Mittra, “A finite element cavity resonance method for waveguide and microstrip line discontinuity problems,” IEEE Trans. Microwave Theory Techn. 42, 433–440 (1994).

    Article  Google Scholar 

  15. A. M. Ivinskaya, A. V. Lavrinenko, and D. M. Shyroki, “Modeling of nanophotonic resonators with the finite-difference frequency-domain method,” IEEE Trans. Antennas Propag. 59, 4155–4161 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. N. Kudryavsev and S. I. Trashkeev, “Formalism of two potentials for the numerical solution of Maxwell’s equations,” Comput. Math. Math. Phys. 53, 1653–1663 (2013).

  17. K. V. Vyaznikov, V. F. Tishkin, and A. P. Favorskii, “Construction of monotone difference schemes of a high approximation order for systems hyperbolic equations,” Mat. Model. 1 (5), 95–120 (1989).

    MathSciNet  MATH  Google Scholar 

  18. A. Harten and S. Osher, “Uniformly high-order accurate nonoscillatory schemes. I” SIAM J. Numer. Anal. 24, 279–309 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  19. X.-D. Liu, S. Osher, and T. Chan, “Weighted essentially non-oscillatory schemes,” J. Comput. Phys. 115, 200–212 (1994).

  20. K. L. Shlager and J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Antennas Propag. Mag. 37, 39–57 (1995).

    Article  Google Scholar 

  21. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).

    Book  Google Scholar 

  22. U. S. Inan and R. A. Marshall, Numerical Electromagnetics. The FDTD Method (Cambridge Univ. Press, Cambridge, 2011).

    Book  Google Scholar 

  23. A. Taflove, S. G. Johnson, and A. Oskooi, Advances in FDTD Computational Electromagnetics: Photonics and Nanotechnology (Artech, London, 2013).

    Google Scholar 

  24. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas. Propag. 14 (3), 302–307 (1966).

    Article  MATH  Google Scholar 

  25. N. N. Kalitkin, “An improved factorization of parabolic schemes,” Dokl. Math. 71, 480–483 (2005).

    MATH  Google Scholar 

  26. A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas. Propag. 39, 1518–1525 (1991).

    Article  Google Scholar 

  27. Zh. O. Dombrovskaya and A. N. Bogolyubov, “Nonmonotonicity of the FDTD scheme in simulation of interfaces between dielectrics,” Uchen. Zapiski Fiz. Facult. Mosk. Univ., No. 4, 1740302 (2017).

  28. G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal, anisotropic dielectrics,” J. Comput. Phys. 226, 1085–1101 (2007).

    Article  MATH  Google Scholar 

  29. A. F. Oskooi, C. Kottke, and S. G. Johnson, “Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing,” Opt. Lett. 34, 2778–2780 (2009).

    Article  Google Scholar 

  30. C. A. Bauer, G. R. Werner, and J. R. Cary, “A second-order 3d electromagnetic algorithm for curved interfaces between anisotropic dielectrics on a Yee mesh,” J. Comput. Phys. 230, 2060–2075 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. R. Werner, C. A. Bauer, and J. R. Cary, “A more accurate, stable FDTD algorithm for electromagnetics in anisotropic dielectrics,” J. Comput. Phys. 255, 436–455 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett. 10, 359–361 (2000).

    Article  Google Scholar 

  33. K. P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wireless Compon. Lett. 11, 158–160 (2001).

    Article  Google Scholar 

  34. R. B. Armenta and C. D. Sarris, “A second-order domain-decomposition method for modeling material interfaces in finite-difference discretizations,” Proceedings of the IEEE/MTT-S International Symposium, 2012, pp. 502–505.

  35. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, London, 2005).

  36. A. A. Samarskii, Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

  37. J. C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  38. R. C. Kirby, A. Logg, and A. R. Terrel, Automated Solution of Differential Equations by the Finite Element Method (Springer, Heidelberg, 2012).

  39. J. S. Hesthaven and D. Gottlieb, “Stable spectral methods for conservation laws on triangles with unstructured grids,” Comput. Methods Appl. Mech. Eng. 175, 361–381 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  40. J. S. Hesthaven, “Spectral penalty methods,' Appl. Numer. Math. 33, 23–41 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  41. J. S. Hesthaven and C. H. Teng, “Stable spectral methods on tetrahedral elements,” SIAM J. Sci. Comput. 21, 2352–2380 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  42. K. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. 49, 749–756 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  43. J. S. Hesthaven and T. Warburton, “High-order nodal methods on unstructured grids. i. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 1–34 (2002).

    Article  MATH  Google Scholar 

  44. S. Piperno and L. Fezoui, “A discontinuous Galerkin FVTD method for 3D Maxwell equations,” CERMICS research report, 2003, p. 4733.

  45. J. S. Hesthaven and T. Warburton, “High order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem,” Phil. Trans. R. Soc. London. A 362, 493–524 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  46. J. S. Hesthaven, “High-order accurate methods in time-domain computational electromagnetics: A review,' Adv. Imaging Eelectron Phys. 127, 59–123 (2003).

  47. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Comput., EMC 23, 377–382 (1981).

    Google Scholar 

  48. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–367 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  49. D. M. Sullivan, “Frequency-dependent FDTD methods using z transforms,” IEEE Trans. Antennas Propag. 40, 1223–1230 (1992).

    Article  Google Scholar 

  50. D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag. 44, 28–34 (1996).

  51. K. Abdijalilov and H. Grebel, “Z-transform theory and FDTD stability,” IEEE Trans. Antennas Propag. 52, 2950–2954 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  52. D. F. Kelley, T. J. Destan, and R. J. Luebbers, “Debye function expansions of complex permittivity using a hybrid particle swarm-least squares optimization approach,” IEEE Trans. Antennas Propag. 55, 1999–2005 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  53. Z. Lin, Y. Fang, J. Hu, and C. Zhang, “On the FDTD formulations for modeling wideband Lorentzian media,” IEEE Trans. Antennas Propag. 59, 1338–1346 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  54. X. T. Dong, N. V. Venkatarayalu, B. Guo, W. Y. Yin, and Y. B. Gan, “General formulation of unconditionally stable ADI-FDTD method in linear dispersive media,” “IEEE Trans. Microwave Theory Techn. 52, 170–174 (2004).

    Article  Google Scholar 

  55. J. L. Young and R. O. Nelson, “A summary and systematic analysis of FDTD algorithms for linearly dispersive media,” IEEE Antennas Propag. Mag. 43, 61–126 (2001).

    Article  Google Scholar 

  56. H. Cai, X. Hu, B. Xiong, and M. S. Zhdanov, “Finite-element time-domain modeling of electromagnetic data in general dispersive medium using adaptive Pade series,” IEEE Antennas Propag. Mag. 109, 194–205 (2017).

    Google Scholar 

  57. Z. Lin, C. Zhang, P. Ou, Y. Jia, and L. Feng, “A generally optimized FDTD model for simulating arbitrary dispersion based on the Maclaurin series expansion, " J. Lightwave Technol. 28, 2843–2850 (2010).

    Article  Google Scholar 

  58. W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, (1997) 401–410.

    Article  Google Scholar 

  59. J. A. Pereda, A. Vegas, and A. Prieto, “FDTD modeling of wave propagation in dispersive media by using the mobius transformation technique,” IEEE Trans. Microwave Theory Techn. 50, 1689–1695 (2002).

    Article  Google Scholar 

  60. J. C. Bolomey, C. Durix, and D. Lesselier, “Time-domain integral equation approach for inhomogeneous and dispersive slab problems,” IEEE Trans. Antennas Propag. AP-26, 658–667 (1978).

    Article  Google Scholar 

  61. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compatibility 32, 222–227 (1990).

    Article  Google Scholar 

  62. R. Luebbers, F. P. Hunsberger, and K. S. Kunz, “A frequency-dependent finite difference time-domain formulation for transient propagation in plasmas,” IEEE Trans. Antennas Propag. 39, 29–43 (1991).

    Article  Google Scholar 

  63. R. Luebbers and F. P. Hunsberger, “FDTD for nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).

    Article  Google Scholar 

  64. I. Giannakis and A. Giannopoulos, “A novel piecewise linear recursive convolution approach for dispersive media using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 62, 2669–2678 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  65. J. L. Young, “Propagation in linear dispersive media: Finite difference time-domain methodologies,” IEEE Trans. Antennas Propag. 43, 422–426 (1995).

    Article  Google Scholar 

  66. T. Kashiwa, N. Yoshida, and I. Fukai, “A treatment by the finite-difference time-domain method of the dispersive characteristics associated with orientation polarization,” Inst. Electron. Inform. Commun. Eng. Trans. E73, 1326–1328 (1990).

    Google Scholar 

  67. T. Kashiwa and I. Fukai, “A treatment by the FD-TD method for the dispersive characteristics associated with electronic polarization,” Microwave Opt. Tech. Lett. 3, 203–205 (1990).

    Article  Google Scholar 

  68. O. P. Gandhi, “A frequency-dependent finite-difference time-domain formulation for general dispersive media,” IEEE Trans. Microwave Theory Tech. 41, 658–665 (1993).

    Article  Google Scholar 

  69. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).

    Article  Google Scholar 

  70. F. Maradei, “A frequency-dependent WETD formulation for dispersive materials,” IEEE Trans. Magnet. 37, 3303–3306 (2001).

    Article  Google Scholar 

  71. G. Kobidze, J. Gao, B. Shanker, and E. Michielssen, “A fast time-domain integral equation based scheme for analyzing scattering from dispersive objects,” IEEE Trans. Magnet. 53, 1215–1226 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  72. X. Zhuansun and X. Ma, “Integral-based exponential time differencing algorithms for general dispersive media and the CFS-PML,” IEEE Trans. Antennas Propag. 60, 3257–3264 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  73. P. G. Petropoulos, “Stability and phase error analysis of FDTD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).

    Article  Google Scholar 

  74. J. L. Young, A. Kittichartphayak, Y. M. Kwok, and D. Sullivan, “On the dispersion errors related to (FD)2/TD type schemes,” IEEE Trans. Microwave Theory Techn. 43, 1902–1910 (1995).

    Article  Google Scholar 

  75. B. Fornberg, “Some numerical techniques for Maxwell’s equations in different types of geometries,” in Topics in Computational Wave Propagation. Lect. Notes Comput. Sci. Eng., vol. 31, 2003, pp. 265–299.

  76. B. Fornberg, J. Zuev, and J. Lee, “Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations,” J. Comput. Appl. Math. 200, 178–192 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  77. S. G. Garcia, T. W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas Wirel. Propag. Lett. 1, 31–34 (2002).

    Article  Google Scholar 

  78. Zh. O. Dombrovskaya and A. N. Bogolyubov, “Improving the accuracy of one-dimensional Yee scheme using grid refinement,” Izv. Ross. Akad. Nauk, Ser. Fiz. 81 (1), 117–120 (2017).

    Google Scholar 

  79. A. I. Tolstykh, Compact Difference Schemes and Their Application in Fluid Dynamics (Nauka, Moscow, 1990).

    Google Scholar 

  80. N. N. Kalitkin and P. V. Koryakin, “Bicompact schemes and layered media,” Dokl. Math. 77, 320–324 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  81. N. N. Kalitkin and P. V. Koryakin, “One- and two-dimensional bicompact schemes in layered media,” Math. Models Comput. Simul. 2, 139–155 (2010).

    Article  MathSciNet  Google Scholar 

  82. N. N. Kalitkin and P. V. Koryakin, Numerical Methods, Vol. 2: Methods of Mathematical Physics (Akademia, Moscow, 2013) [in Russian].

  83. A. N. Tikhonov and A. A. Samarskii, “On the convergence of difference schemes in the class of discontinuous coefficients,” Dokl. Akad. Nauk SSSR 8, 529–532 (1959).

    MathSciNet  MATH  Google Scholar 

  84. A. A. Samarskii, Introduction to the Theory of Finite Difference Schemes (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  85. A. A. Samarskii and Yu. P. Popov, Difference Methods for Solving Fluid Dynamics Problems (Nauka, Moscow, 1992) [in Russian].

    Google Scholar 

  86. A. G. Sveshnikov, “Principles of radiation,” Dokl. Akad. Nauk SSSR 3, 517–520 (1950).

    MathSciNet  Google Scholar 

  87. L. F. Richardson and J. A. Gaunt, “The deferred approach to the limit,” Phil. Trans. A 226, 299–349 (1927).

    MATH  Google Scholar 

  88. G. I. Marchuk, Splitting Methods (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  89. N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Computations on Quasiuniform Grids (Fizmatlit, Moscow, 2005) [in Russian].

  90. Z. Meglicki, S. K. Gray, and B. Norris, “Multigrid FDTD with chombo,” Comput. Phys. Commun. 176, 109–120 (2007).

    Article  MATH  Google Scholar 

  91. A. Van Londersele, D. De Zutter, and D. V. Ginste, “An in-depth stability analysis of nonuniform FDTD combined with novel local implicitization techniques,” J. Comput. Phys. 342, 177–193 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  92. D. S. Balsara and J. J. Simpson, “Making a synthesis of FDTD and DGTD schemes for computational electromagnetics,” IEEE J. Multiscale Multiphys. Comput. Techniques 5, 99 (2020).

    Article  Google Scholar 

  93. V. S. Ryaben’kii and A. F. Filippov, On Stability of Difference Equations (Gos. Iz-vo Tekhniko-Teoreticheskoi Literatury, Moscow, 1956) [in Russian].

    Google Scholar 

Download references

ACKNOWLEDGMENTS

We are grateful to N.N. Kalitkin for valuable remarks and discussions and to A.N. Bogolyubov for attention to this work.

Funding

This work was supported by the Russian Science Foundation, project no. 20-71-00097.

Author information

Authors and Affiliations

  1. Moscow State University, 119991, Moscow, Russia

    A. A. Belov & Zh. O. Dombrovskaya

  2. Peoples’ Friendship University of Russia (RUDN University), 117198, Moscow, Russia

    A. A. Belov

Authors
  1. A. A. Belov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zh. O. Dombrovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. A. Belov or Zh. O. Dombrovskaya.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belov, A.A., Dombrovskaya, Z.O. Highly Accurate Methods for Solving One-Dimensional Maxwell Equations in Stratified Media. Comput. Math. and Math. Phys. 62, 84–97 (2022). https://doi.org/10.1134/S0965542522010043

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542522010043

Keywords:

Search

Navigation