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

Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves
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Year:    2016

Author:    Ya Zhang, Duc Duy Nguyen, Kewei Du, Jin Xu, Shan Zhao

Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 3 : pp. 353–385

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 the other 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 Galerkin Maxwell 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.2014.m811

Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 3 : pp. 353–385

Published online:    2016-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    33

Keywords:    Maxwell's equations finite-difference time-domain (FDTD) discontinuous Galerkin time-domain (DGTD) transverse magnetic (TM) and transverse electric (TE) systems high order interface treatments matched interface and boundary (MIB).

Author Details

Ya Zhang

Duc Duy Nguyen

Kewei Du

Jin Xu

Shan Zhao

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