TY - JOUR
T1 - Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves
AU - Zhang , Ya
AU - Nguyen , Duc Duy
AU - Du , Kewei
AU - Xu , Jin
AU - Zhao , Shan
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 353
EP - 385
PY - 2018
DA - 2018/05
SN - 8
DO - http://doi.org/10.4208/aamm.2014.m811
UR - https://global-sci.org/intro/article_detail/aamm/12093.html
KW - 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).
AB - <p style="text-align: justify;">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&#39;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.</p>