Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T12:32:06.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves

Part of: Optics, electromagnetic theory - Basic methods Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  27 January 2016

Ya Zhang ,
Duc Duy Nguyen ,
Kewei Du ,
Jin Xu  and
Shan Zhao
Ya Zhang
Affiliation:
Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
Duc Duy Nguyen
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
Kewei Du
Affiliation:
Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
Jin Xu
Affiliation:
Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
Shan Zhao*
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
*
*Corresponding author. Email:szhao@ua.edu (S. Zhao)
Get access

Abstract.

This paper is devoted to time domain numerical solutions of two-dimensional (2D) material interface problems governed by the transverse magnetic (TM) and transverse electric (TE) Maxwell's equations with discontinuous electromagnetic solutions. Due to the discontinuity in wave solutions across the interface, the usual numerical methods will converge slowly or even fail to converge. This calls for the development of advanced interface treatments for popular Maxwell solvers. We will investigate such interface treatments by considering two typical Maxwell solvers – one based on collocation formulation and another based on Galerkin formulation. To restore the accuracy reduction of the collocation finite-difference time-domain (FDTD) algorithm near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed in this work, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. The resulting MIB time-domain (MIBTD) scheme satisfies the jump conditions locally and suppresses the staircase approximation errors completely over the Yee lattices. In the discontinuous Galerkin time-domain (DGTD) algorithm – a popular GalerkinMaxwell solver, a proper numerical flux can be designed to accurately capture the jumps in the electromagnetic waves across the interface and automatically preserves the discontinuity in the explicit time integration. The DGTD solution to Maxwell interface problems is explored in this work, by considering a nodal based high order discontinuous Galerkin method. In benchmark TM and TE tests with analytical solutions, both MIBTD and DGTD schemes achieve the second order of accuracy in solving circular interfaces. In comparison, the numerical convergence of the MIBTD method is slightly more uniform, while the DGTD method is more flexible and robust.

Keywords

Maxwell's equationsfinite-difference time-domain (FDTD)discontinuous Galerkin time-domain (DGTD)transverse magnetic (TM) and transverse electric (TE) systemshigh order interface treatmentsmatched interface and boundary (MIB)

MSC classification

Secondary: 65M06: Finite difference methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 78M10: Finite element methods 78M20: Finite difference methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 3 , June 2016 , pp. 353 - 385
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley & Sons, 1989.Google Scholar
[2]Bossavit, A. and Verite, J. C., A mixed FEM-BIEM method to solve 3-D eddy current problems, IEEE Trans. Magn., 18 (1982), pp. 431–435.Google Scholar
[3]Cai, W. and Deng, S., An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case, J. Comput. Phys., 190 (2003), pp. 159–183.Google Scholar
[4]Cai, W., Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering and Electron Transport, Cambridge University Press, Cambridge, 2012.Google Scholar
[5]Chen, M.-H., Cockburn, B. and Reitich, F., High-order RKDG methods for computational electromagnetics, J. Sci. Comput., 22 (2005), pp. 205–226.Google Scholar
[6]Cockburn, B., Li, F. and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194 (2004), pp. 588–610.CrossRefGoogle Scholar
[7]Cohen, G., Ferrieres, X. and Pernet, S., A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in the time domain, J. Comput. Phys., 217 (2006), pp. 340–363.Google Scholar
[8]Ditkowski, A., Dridi, K. and Hesthaven, J. S., Convergent Cartesian grid methods for Maxwell's equations in complex geometries, J. Comput. Phys., 170 (2001), pp. 39–80.Google Scholar
[9]Donderici, B. and Teixeira, F. L., Mixed finite-element time-domain method for Maxwell equations in doubly-dispersive media, IEEE Trans.Microw. Theory Tech., 56 (2008), pp. 113–120.Google Scholar
[10]Gedney, S. D., Luo, C., Roden, J. A., Crawford, R. D., Guernsey, B., Miller, J. A., Kramer, T. and Lucas, E. W., The discontinuous Galerkin finite-element time-domain method solution of Maxwell's equations, J. Appl. Comput. Electromagn. Soc., 24 (2009), pp. 129–142.Google Scholar
[11]Grote, M., Schneebeli, A. and Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell's equations: energy norm error estimates, J. Comput. Appl. Math., 204 (2007), pp. 375–386.Google Scholar
[12]Hesthaven, J. S., High-order accurate methods in time-domain computational electromagnetics, A review, Adv. Imag. Electron Phys., 127 (2003), pp. 59–123.Google Scholar
[13]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Springer, New York, 2008.Google Scholar
[14]Huang, Y. and Li, J., Interior penalty discontinuous Galerkin method for Maxwell's equation in cold plasma, J. Sci. Comput. 41 (2009), pp. 321–340.Google Scholar
[15]Huang, Y., Li, J. and Yang, W., Interior penalty DG methods for Maxwell's equations in dispersive media, J. Comput. Phys., 230 (2011), pp. 4559–4570.CrossRefGoogle Scholar
[16]Huynh, L. N. T., Nguyen, N. C., Peraire, J. and Khoo, B. C., A high-order hybridizable dis- continuous Galerkin method for elliptic interfac problems, Int. J. Numer. Method Eng., 93 (2013), pp. 183–200.Google Scholar
[17]Ji, X., Cai, W. and Zhang, P. W., High-order DGTD method for dispersive Maxwell's equations and modelling of silver nanowire coupling, Int. J. Numer. Methods Eng., 69 (2007), pp. 308–325.Google Scholar
[18]Jiao, D. and Jin, J. M., Time-domain finite-element modeling of dispersive media, IEEE Microw. Wireless Comput. Lett., 11 (2001), pp. 220–222.Google Scholar
[19]Li, J., Posteriori error estimation for an interiori penalty discontinuous Galerkin method for Maxwell's equations in cold plasma, Adv. Appl. Math. Mech., 1 (2009), pp. 107–124.Google Scholar
[20]Li, J., Development of discontinuous Galerkin methods for Maxwell's equations in metamaterials and perfectly matched layers, J. Comput. Appl. Math., 236 (2011), pp. 950–961.Google Scholar
[21]Li, J. and Huang, Y., Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials, Springer, Berlin, 2013.Google Scholar
[22]Li, J. and Hesthaven, J. S., Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258 (2014), pp. 915–930.Google Scholar
[23]Li, Ping, Shi, Yifei, Jiang, Lijun and Hakan Bağgci, , A hybrid time-domain discontinuous Galerkin-boundary integral method for electromagnetic scattering analysis, IEEE Trans. Antennas Propagation, 62 (2014), pp. 2841–2846.Google Scholar
[24]Lu, T., Zhang, P. W. and Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions, J. Comput. Phys., 200 (2004), pp. 549–580.CrossRefGoogle Scholar
[25]Lu, T., Zhang, P. W. and Cai, W., Discontinuous Galerkin time domain method for GPR simu- lation in dispersive media, IEEE Trans. Geosci. Remote Sensing, 43 (2005), pp. 72–80.Google Scholar
[26]Monk, P. and Richter, G. R., A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Comput., 22 (2005), pp. 443–477.Google Scholar
[27]Montseny, E., Pernet, S., Ferriéres, X. and Cohen, G., Dissipative terms and local time- stepping improvements in a spatial high order discontinuous Galerkin scheme for the time-domain Maxwell's equations, J. Comput. Phys., 227 (2008), pp. 6795–6820.CrossRefGoogle Scholar
[28]Mur, G. and De Hoop, A. T., A finite-element method for computing three-dimensional electro- magnetic fields in inhomogeneous media, IEEE Trans. Magn. 21 (1985), pp. 2188–2191.Google Scholar
[29]Nguyen, D. D. and Zhao, S., High order FDTD methods for transverse magnetic modes with dispersive interfaces, Appl. Math. Comput., 226 (2014), pp. 699–707.Google Scholar
[30]Nguyen, D. D. and Zhao, S., Time-domain matched interface and boundary (MIB) modeling of Debye dispersive media with curved interfaces, J. Comput. Phys., 278 (2014), pp. 298–325.Google Scholar
[31]Nédélec, J. C., Mixed finite elements in R3, Numer. Meth., 35 (1980), pp. 315–341.CrossRefGoogle Scholar
[32]Nédélec, J. C., A new family of mixed finite elements in R3, Numer.Math., 50 (1986), pp. 57–81.Google Scholar
[33]Stoykov, N. S., Kuiken, T. A., Lowery, M. M. and Taflove, A., Finite-element time-domain algorithms for modeling linear Debye and Lorentz dielectric dispersions at low frequencies, IEEE Trans. Biomed. Eng., 50 (2003), pp. 1100–1107CrossRefGoogle ScholarPubMed
[34]Taflove, A. and Hagness, S. C., Computational Electrodynamics: The Finite-Time Domain-Method, 3rd ed., Norwood, MA: Artech House, 2005.Google Scholar
[35]Wang, B., Xie, Z. and Zhang, Z., Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys., 229 (2010), pp. 8552–8563.Google Scholar
[36]Xie, Z. Q., Chan, C.-H. and Zhang, B., An explicit fourth order staggered finite-difference time-domain method for Maxwell's equations, J. Comput. Appl. Math., 147 (2002), pp. 75–98.CrossRefGoogle Scholar
[37]Xie, Z. Q., Chan, C.-H. and Zhang, B., An explicit fourth order orthogonal curvilinear staggered-grid FDTD method for Maxwell's equations, J. Comput. Phys., 175 (2002), pp. 739–763.Google Scholar
[38]Xu, J., Mustapha, B., Ostroumov, P. N. and Nolen, J., Solving 2D2V Vlasov equation with high order spectral element method, Commun. Comput. Phys., 8 (2010), pp. 159–184.Google Scholar
[39]Xu, J., Ostroumov, P. N., Nolen, J. and Kim, K. J., Scalable direct Vlasov solver with discontinuous Galerkin method on unstructured mesh, SIAM J. Sci. Computing, 32 (2010), pp. 3476–3494.Google Scholar
[40]Yefet, A. and Turkel, E., Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients, Appl. Numer. Math., 33 (2000), pp. 125–134.CrossRefGoogle Scholar
[41]Yefet, A. and Petropoulos, P. G., A staggered Fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations, J. Comput. Phys., 168 (2001), pp. 286–315.Google Scholar
[42]Zhao, S. and Wei, G. W., High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces, J. Comput. Phys., 200 (2004), pp. 60–103.Google Scholar
[43]Zhao, S., High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces, J. Comput. Phys., 229 (2010), pp. 3155–3170.CrossRefGoogle Scholar
[44]Zhao, S., High order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media, Opt. Lett., 36 (2011), pp. 3245–3247.CrossRefGoogle ScholarPubMed
[45]Zhao, S., A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces, J. Sci. Comput., 63 (2015), pp. 118–137.CrossRefGoogle Scholar
[46]Zhang, C. M. and Leveque, R. J., The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion, 25 (1997), pp. 237–263.Google Scholar
[47]Zhou, Y. C., Zhao, S., Feig, M. and Wei, G. W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213 (2006), pp. 1–30.Google Scholar