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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)
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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 

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