扩展的六阶段分步FDTD方法（包括集总电容器）的稳定性
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Stability of the Extended Six-Stage Split-Step FDTD
Method Including Lumped Capacitors
Yong-Dan Kong
1, 2
1
School of Electronic and Information Engineering,
South China University of Technology,
Guangzhou, Guangdong 510640, China.
Email: eeydkong@scut.edu.cn
Qing-Xin Chu
1, 2
2
The State Key Laboratory of Millimeter Waves,
Southeast University,
Nanjing, Jiangsu 210096, China.
Email: qxchu@scut.edu.cn
Abstract—The stability of the six-stage split-step finite-
difference time-domain (SS-FDTD) method including lumped
capacitors is studied. In particular, the implicit different
formulation for the lumped capacitor is analyzed. Then, its
unconditional stability is analytically proven by combining the
von Neumann method with the Jury criterion. Finally, its
unconditional stability is verified through numerical experiments.
Keywords—Finite-difference time domain (FDTD); extended
method; split-step scheme; lumped capacitors; numerical stability.
I. INTRODUCTION
Recently, to remove the Courant–Friedrichs–Lewy (CFL)
limitation on the time-step size of the finite-difference time-
domain (FDTD) method [1], unconditionally-stable methods
such as alternating direction implicit (ADI) [2], split-step (SS)
[3, 4], locally one-dimensional (LOD) [5], and leapfrog ADI
[6] FDTD methods were developed. Moreover, an
unconditionally-stable FDTD method with six sub-steps was
presented in [7], which has simpler procedure formulation.
Along another line, lumped elements have been successfully
incorporated into the conventional FDTD [8] and other
unconditionally stable FDTD methods [9-12]. However, the
six-stage SS-FDTD method has not been extended to include
lumped elements; then the study on its stability is very useful.
The stability of the extended six-stage SS-FDTD method
including lumped capacitors is studied in this paper. First, the
formulation of the extended six-stage SS-FDTD method is
given. Second, the numerical stability of the proposed method
including lumped capacitors is analyzed by combining the von
Neumann method with the Jury criterion, and the results show
that the proposed method is unconditionally stable. Finally, a
numerical experiment is simulated to demonstrate the validity
of the theoretical results.
II. F
ORMULATION
Without loss of generality, assume that the lumped
capacitor is placed along the +z direction and located in a
lossless region with the electric permittivity İ and magnetic
permeability ȝ. The contribution of the lumped capacitor is
presented by
Lz
J
JG
. The formulation of the extended six-stage
SS-FDTD method including lumped capacitors is shown as
follows.
[]
.
Lz
ut MuJ∂∂= −
ε
GGJG
(1)
where
[, ,, , , ]
T
xyz x y z
uEEEHHH=
G
, and [M] is the Maxwell’s
matrix. According to the positive and negative of the
x, y, and
z coordinate directions, the Maxwell’s matrix is split into six
sub-matrices, [
D
1
], [E
1
], [F
1
], [D
2
], [E
2
], and [F
2
]. Then, (1)
can be written as
[
]
[
]
[
]
[
]
[
]
[
]
111 2 22
Lz
ut Du Eu Fu Du Eu FuJ
ε
∂∂= + + + + + −
GGGGGGGJG
. (2)
Due to the limitation of space, [
M], [D
1
], [E
1
], [F
1
], [D
2
], [E
2
],
and [
F
2
] are not shown here, they can be found in [7].
By using the split-step scheme [3], (2) is divided into six
sub-equations, from
n to n+1, one time step is divided into six
sub-steps accordingly,
nĺn+1/6, n+1/6ĺn+2/6,
n+2/6ĺn+3/6, n+3/6ĺn+4/6, n+4/6ĺn+5/6, and
n+5/6ĺn+1. Furthermore, by using the Crank-Nicolson
scheme [7], the right side of the equations can be
approximated. Six sub-procedures are generated as follows
[] []
()
[] []
()
1/6 1/12
11
22
nnn
Lz
I t Du I t Du bJ
++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3a)
[] []
()
[] []
()
2/6 1/6 1/4
11
22
nnn
Lz
I t Eu I t Eu bJ
+++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3b)
[] []
()
[] []
()
3/6 2/6 5/12
11
22
nnn
Lz
ItFu ItFu bJ
+++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3c)
[] []
()
[] []
()
4/6 3/6 7/12
22
22
nnn
Lz
I t Du I t Du bJ
+++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3d)
[] []
()
[] []
()
5/6 4/6 3/4
22
22
nnn
Lz
I t Eu I t Eu bJ
+++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3e)
[] []
()
[] []
()
15/611/12
22
22 .
nnn
Lz
I t Fu I t Fu bJ
+++
−Δ ⋅ = +Δ ⋅ − ⋅
GGJG
(3f)
For the sub-step 1,
E
z
n+1/6
and H
y
n+1/6
in (3a) can be
rewritten as
()
()
1/6 1/6 1/12nn nn n
zz yy Lz
EEbHHxbxyI
++ +
=+⋅∂ + ∂−ΔΔ⋅
(4a)
()
1/6 1/6
.
nn nn
yy zz
H
Hd E E x
++
=+⋅∂ + ∂
(4b)
where
b=ǻt/(2İ), d = ǻt/(2ȝ).
Here, we only consider the implicit difference scheme for
the lumped capacitor. According to the implicit scheme, the
voltage and current characteristic of the lumped capacitor at
the time step of
t = (n+1/12) ǻt is:
()
()
112 112 16
6
nn nn
Lz z z z
ICVtCztEE
++ +
=⋅∂ ∂= Δ Δ⋅ −
// /
. (5)
III. N
UMERICAL STABILITY ANALYSIS
The stability of the extended six-stage SS-FDTD method
including lumped capacitors is analyzed by combining the von
Neumann method with the Jury criterion. This analyzing
procedure has also been applied to the stability analysis of the
439
978-1-4799-8897-6/15/$31.00
c
2015 IEEE
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