-
A comparative study on Explicit and Implicit FDTDmethods for
Electromagnetic simulation
Gurinder Singh1, R. S. Kshetrimayum2 and Thingbaijam Rajkumari
Chanu31,3Department of Electronics and Communication Engineering,
NIT Mizoram, Aizawl, India
[email protected] and [email protected] of
Electronics and Electrical Engineering, IIT Guwahati, India
[email protected]
Abstract—In this paper, Explicit and Implicit FDTD methodshave
been compared in terms of required computational timefor various
problem space. The unconditionally stable Implicitmethod viz. Crank
Nicolson (CN) and Alternating DirectionImplicit (ADI) FDTD have
been used for analysis of free spacewave propagation simulation.
Also, the Crank Nicolson FDTDmethod has been used for the analysis
of Lorentzian DoubleNegative (DNG) metamaterial. There can be
significant reductionin simulation time with increase in time step
size and decrease innumber of required time steps for implicit
methods as comparedto conventional explicit FDTD method. It is
expected that theseimplicit methods should be stable for any time
step size, howeverthere is always an upper bound on maximum time
step size inorder to maintain desired level of numerical accuracy
in resultfor a particular problem space.
Index Terms—Crank Nicolson, Alternating Direction
Implicit,Auxiliary Differential Equation (ADE), Double Negative
Meta-material.
I. INTRODUCTION
The unconditionally stable methods viz. Crank Nicolson(CN) and
Alternating Direction Implicit (ADI) methods canreduce simulation
time for a larger problem space. The con-ventional explicit method
has a major drawback that there isalways limit on maximum time step
size in order to get a stableFDTD solutions. To ovecome this
limitation, Crank Nicolsonand Alternating Direction Implicit FDTD
methods are usedwhich have been discussed in next section.
A. 1D Crank Nicolson FDTD method
Maxwell’s Equation in 1-D (E is in the y-direction, H isin the
z-direction and propagation is in the x-direction and ofcourse in
time t) is
µ∂Hz∂t
= −∂Ey∂x
(1)
and
�∂Ey∂t
= −∂Hz∂x− σEy (2)
From the definition of Crank Nicolson method, above equa-tions
are then modified as
µ∂Hz∂t
= −12
(∂En+1y∂x
+∂Eny∂x
) (3)
and
�∂Ey∂t
= −12
(∂Hn+1z∂x
+∂Hnz∂x
)− σEny (4)
The equations are then discretized starting with
temporaldiscretization and saving the spatial discretization for
later asshown below:
µHn+1z −Hnz
∆t= −1
2(∂En+1y∂x
+∂Eny∂x
) (5)
Hn+1z = Hnz −
∆t
2µ(∂En+1y∂x
+∂Eny∂x
) (6)
�En+1y − Eyn
∆t= −1
2(∂Hn+1z∂x
+∂Hnz∂x
)− σEny (7)
En+1y = Eny −
∆t
2�(∂Hn+1z∂x
+∂Hnz∂x
)− ∆t�σEny (8)
By rearranging, we have
En+1y = Eny −
∆t
2�(∂(Hnz − ∆t2µ (
∂En+1y∂x +
∂Eny∂x ))
∂x+∂Hnz∂x
)
(9)
−∆t�σEny
En+1y = Eny −
∆t
2�
∂Hnz∂x
+∆t2
4µ�
∂2En+1y∂x2
+∆t2
4µ�
∂2Eny∂x2
(10)
−∆t2�
∂Hnz∂x− ∆t
�σEny
En+1y −∆t2
4µ�
∂2En+1y∂x2
= Eny −∆t
2�
∂Hnz∂x
+∆t2
4µ�
∂2Eny∂x2
(11)
−∆t2�
∂Hnz∂x− ∆t
�σEny
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The above equation is now discretized spatially,
En+1i −∆t2
4µ�∆x2(En+1i+1 − 2E
n+1i + E
n+1i−1 ) = E
ni −
∆tσ
�Eni
(12)
+∆t2
4µ�∆x2(Eni+1 − 2Eni + Eni−1)−
∆t
∆x�(Hni −Hni−1),
En+1i −p
2(En+1i+1 − 2E
n+1i + E
n+1i−1 ) = (1− q)E
ni
(13)
+p
2(Eni+1 − 2Eni + Eni−1)− r(Hni −Hni−1)
For simplification, the following substitutions are made:
p= ∆t2
2µ�∆x2 ,q= ∆tσ� ,r= ∆t∆x�
The value of electric field variation at (n+ 1)th time step
interms of p, q and r is given as
−p2En+1i−1 + (1 + p)E
n+1i −
p
2En+1i−1 =
p
2Eni−1 (14)
+(1− p− q)Eni +p
2Eni +
p
2Eni+1 − r(Hni −Hni−1).
In matrix form, our Crank Nicolson implementation forsolving for
E vector in 1-D looks like this:
(I + pG)En+1 = ((1− q)I− pG)En − (rJ)Hn (15)
where I, G and J are square matrices of the same order m,where m
points exist in the FDTD space. The matrix I is theidentity matrix
of order m. The matrices G, J, E and H canbe expressed as:
G=
1 − 12− 12 1 −
12
− 12 1. . .
. . . . . . . . .− 12 1 −
12
− 12 1
J=
1−1 1
−1 1. . . . . .
−1 1−1 1
En+1=
En+11En+12En+13
...En+1m
and Hn=
Hn1Hn2Hn3
...Hnm
B. 2D Alternating Direction Implicit (ADI) FDTD
The Alternating Direction Implicit (ADI) method breakstime step
into two sub-iteration: advancing fields from nth to
(n+ 1/2)th and secondly from (n+ 1/2)th to (n+ 1)th
timestep.First Procedure:
Hn+ 12x (i, j +
1
2) = Hnx (i, j +
1
2) (16)
− ∆t2µ∆y
[Enz (i, j + 1)− Enz (i, j)]
Hn+ 12y (i+
1
2, j) = Hny (i+
1
2, j) (17)
− ∆t2µ∆x
[En+ 12z (i+ 1, j)− E
n+ 12z (i, j)]
En+ 12z (i, j) = E
nz (i, j) (18)
+∆t
2�∆x[H
n+ 12y (i+
1
2, j)−Hn+
12
y (i−1
2, j)]
− ∆t2�∆y
[Hnx (i, j +1
2)−Hnx (i, j −
1
2)]
Second Procedure:
Hn+1x (i, j +1
2) = H
n+ 12x (i, j +
1
2) (19)
− ∆t2µ∆y
[En+1z (i, j + 1)− En+1z (i, j)]
Hn+1y (i+1
2, j) = H
n+ 12y (i+
1
2, j) (20)
− ∆t2µ∆x
[En+ 12z (i+ 1, j)− E
n+ 12z (i, j)]
En+1z (i, j) = En+ 12z (i, j) (21)
+∆t
2�∆x[H
n+ 12y (i+
1
2, j)−Hn+
12
y (i−1
2, j)]
− ∆t2�∆y
[Hn+1x (i, j +1
2)−Hn+1x (i, j −
1
2)]
The first sub-iteration equation (17) is substituted
intoequation (18) to eliminate Hn+
12
y term so that En+ 12z can be
found by equation (22). The equation Hn+12
y can then besolved by equation (17). Similiarly in the second
procedure,equation (19) is substituted into (21) to eleminate Hn+1x
termso that En+1z can be found by equation (26). Again H
n+1x
can be found from equation (19).
αEn+ 12z (i− 1, j) + βE
n+ 12z (i, j) + γE
n+ 12z (i+ 1, j) (22)
= Enz +∆t
2�∆x[Hny (i+
1
2, j)−Hny (i−
1
2, j)]
− ∆t2�∆x
[Hnx (i, j +1
2)−Hnx (i, j −
1
2)]
where,
α = − ∆t2µ∆x
.∆t
2�∆x(23)
γ = − ∆t2µ∆x
.∆t
2�∆x(24)
β = 1− α− γ. (25)
-
αEn+1z (i, j − 1) + βEn+1z (i, j) + γEn+1z (i, j + 1) (26)
= En+ 12z +
∆t
2�∆y[H
n+ 12y (i+
1
2, j)−Hn+
12
y (i−1
2, j)]
− ∆t2�∆x
[Hn+ 12x (i, j +
1
2)−Hn+
12
x (i, j −1
2)]
II. ONE DIMENSIONAL FREE SPACE SIMULATION WITHCRANK NICOLSON
(CN) FDTD IMPLICIT METHOD
Pure Crank Nicolson implicit scheme was used for onedimensional
free space simulation. Murs ABC was used totruncate the problem
space. A Gaussian pulse that originatesin the centre propagates
outward and is absorbed withoutreflecting back in the problem
space. A comparison was madebetween conventional explicit FDTD and
Implicit CN-FDTDin terms of computational effieciency. In order to
prove that thebetter computational effieciency acheived by the
CN-FDTDscheme, the MATLAB programs for both schemes were run.The
Fig. 1 was obtained when the time step size ∆t used forboth
conventional FDTD and CN-FDTD are same.It can be observed from the
table I that the CNFDTD schemeachieves a considerable reduction of
simulation time andthe results from the CN-FDTD scheme are accurate
enough,although there is some accuracy degradation compared withthe
ones from the conventional FDTD scheme.
Fig. 1: Electric field variation Ex for one dimensional
freespace propagation simulation using Crank Nicolson methodand
conventional FDTD at n=200 with Murs ABC
Fig. 2: Electric field variation Ex for one dimensional
freespace propagation simulation using Crank Nicoloson methodand
conventional FDTD at n=145 and n=200 respectively withMur’s ABC for
relative time step size of ∆tCNFDTD∆tFDTD = 1.5.
TABLE I: Comparison of computational time for ExplicitFDTD and
CN FDTD. Here, r=∆tFDTD.
Method Time step Step number Total time used(s)Explicit FDTD
62.8ps 200 3.333
CNFDTD 94.2ps (1.5r) 145 2.58CNFDTD 125.6ps (2r) 118 2.098
III. TWO DIMENSIONAL FREE SPACE SIMULATION WITHALTERNATING
DIRECTION IMPLICIT (ADI) METHOD
A Gaussian pulse was initiated in the centre across 100x 100
grid. There was no perfectly matched Layer (PML)implementation. The
cell size and number of step sizes weretaken to be 1cm and 90
respectively. The simulation resultshave been shown in Fig. 3 at
different time instants. Thewave behaviour at a point of
observation (x0=40,y0=40) wasanalyzed for both conventional FDTD
and ADI-FDTD. It canbe seen from the Fig. 4 that at point of
observation (x0=40,y0=40), the electric field early time response
is approximatelysame for both the methods as expected.The table II
suggests that with increase in time step size and
(a) (b)
(c) (d)
Fig. 3: Results of a simulation using 2D ADI-FDTD: A Gaus-sian
pulse is initiated in the middle and travels outward across100× 100
cells. (a)T=25, (b)T=50, (c)T=75 and (d)T=90.
Fig. 4: Electric field variation at a point of observation
(xo=40,yo=40) for both conventional explicit FDTD and ADI-FDTD
-
decrease in number of required time steps, there is
significantreduction in simulation time in case of ADI-FDTD as
com-pared to conventional FDTD with fair amount of
numericalaccuracy.
TABLE II: Comparison of computational time for ExplicitFDTD and
ADI FDTD. Here, r=∆tFDTD.
Method Time step Step number Total time used(s)Explicit FDTD
16.67ps 90 19.73
ADIFDTD 25.05ps (1.5r) 70 16.43
IV. CRANK NICOLSON BASED FDTD ANALYSIS OFLORENTZIAN DNG
METAMATERIAL
Wave propagation in dispersive materials with
simultaneousnegative permeability, µ and permittivity, � known as
DNGmetamaterial can be analyzed using FDTD methods by meansof
Lorentz model and Auxiliary Differential Equation (ADE)method which
is explained in next section.
A. Lorentz model
In the Lorentzian model, the frequency-dependence of elec-tric
and magnetic susceptibility functions are given as
χe,L(ω) =ω2pe
ω2oe − ω2 + jτeω(27)
χm,L(ω) =ω2pm
ω2om − ω2 + jτmω(28)
where ωpe and ωpm are the plasma frequencies, ωoe and ωomare the
resonance frequencies, and τe and τm are the dampingcoefficients
respectively.
B. Auxiliary Differential Equation representation for
Consti-tutive Equation
The Auxiliary Differential Equation method converts
thefrequency-domain equation of Lorentzian model into a timedomain
differential equation using jω ←→ ∂∂t and −ω
2 ←→∂2
∂t2 . This yields
D(ω) = �0�∞E(ω) + Sk(ω), (29)B(ω) = µ0µ∞H(ω) + Jk(ω), (30)
where,
Sk(ω) = �0ω2pe
ω2oe − ω2 + jτeωE(ω) (31)
Jk(ω) = µ0ω2pm
ω2om − ω2 + jτmωH(ω) (32)
The inverse Fourier transforms of these two are
∂2Sk(t)
∂t2+ τe
∂Sk(t)
∂t+ ω20eSk(t) = �0ω
2peE(t) (33)
∂2Jk(t)
∂t2+ τm
∂Jk(t)
∂t+ ω20mJk(t) = µ0ω
2pmH(t) (34)
The discrete form of above equation can be given as
Sn+1k = [2−∆t2ω20e1 + 0.5∆tτe
]Snk + [0.5∆tτe − 10.5∆tτe + 1
]Sn−1k (35)
+[∆t2�0ω
2pe
1 + 0.5∆tτe]En
Jn+1k = [2−∆t2ω20m1 + 0.5∆tτm
]Jnk + [0.5∆tτm − 10.5∆tτm + 1
]Jn−1k (36)
+[∆t2µ0ω
2pm
1 + 0.5∆tτm]Hn
These can then be used in iterative form and FDTD loops canbe
updated accordingly:
Dn+1(i) = Dn(i) +∆t
∆x[Hn(i)−Hn(i− 1)], (37)
Bn+1(i) = Bn(i) +∆t
∆x[En(i+ 1)− En(i)], (38)
En+1 =Dn+1 − Sn+1k
�0�∞(39)
Hn+1 =Bn+1 − Jn+1k
µ0µ∞(40)
Fig. 5: The flowchart of the Auxiliary Differential EquationFDTD
procedure [6]. Here, MTM represents Metamaterial.
C. Simulation Details and Result
Numerical simulation results from ADE method describedin the
previous section are presented for the one-dimensionalFDTD case for
a Lorentz material with the parameters foe =fom =0.1591GHz, fpe =
fpm = 1.1027 GHz, and a dampingfactor τ =1× 108 rad/s.The frequency
of operation was set to4.71×109 rad/s in order to yield a
refractive index of n ≈ -1.The problem space was taken to be 200
cells. The cell size wastaken to be λ/10 with operating frequency
of 0.796 GHz. Thecorresponding time step was calculated using ∆t =
0.5∆x/c,where c is the speed of light. The metamaterial slab
extendedfrom cell 60 to cell 90. Outside this range was free
space.First-order Mur-type absorbing blocks were located at
bothends to turncate the problem space. A sinusoidal source was
-
launched in the free-space region at a node 5 cells from theleft
boundary.It is evident from Fig. 6 that wave amplitudes in the
DNGmetamaterial are much higher than that of the normal freespace
(DPS region). So, the DNG material thus enhancesthe energy or
intensity of the wave at region, thereby ab-sorbing maximum energy
from the surrounding mediumshence preserves the energy. The table
III suggests that theCrank Nicolson based FDTD analysis of
Lorentzian DNGmetamaterial requires comparatively less simulation
time ascompared to conventional FDTD with increase in time stepsize
and decrease in number of required step number.
Fig. 6: Electric field variation for one dimensional
FDTDanalysis of Lorentzian DNG metamaterial using Pure
CrankNicolson scheme at n=460 for relative time step size
of∆tCNFDTD
∆tFDTD= 1.5.
TABLE III: Comparison of computational time and error
forExplicit FDTD and CNFDTD for one dimensional FDTDanalysis of
Lorentzian DNG metamaterial. Here, r=∆tFDTD.
Method Time step Step number Simulation Time(s)Explicit FDTD
62.8ps 620 53.624
CNFDTD 94.2ps(=1.5r) 460 32.598CNFDTD 125.6ps(=2r) 380
27.149
V. TIME STEP SIZE LIMITATION
Any unconditionally stable implicit method is stable for anytime
step size, however the time step size is still limited be-cause of
numerical accuracy required. In practice, to maintaina reasonable
accuracy, the time step size should be chosensuch that the
resulting Courant number is much smaller thanthe mesh density, N.
Fig. 7 shows the usable courant numbersat different values of mesh
densities for accuracies of 99%,95%, 85% and 65% respectively for
one dimensional analysisof DNG metamaterial using Crank Nicolson
implicit method.
It can be stressed here that the Crank-Nicolson implicitmethods
do have an upper bound limit to the time-step size.This limit does
not come from the stability requirement as inYees FDTD, but from a
sampling limit which is more stricterthan the Nyquist limit [8]. In
reality, the meaningful allowable
Fig. 7: Relationship of usable courant number with mesh
den-sity, N= λ∆x for one dimensional analysis of DNG
metamaterialusing Crank Nicolson implicit method
time-step size to be used in CN-FDTD should be determinedfrom
the desired numerical accuracy, which increases whenthe usable
Courant number becomes more and more smaller.Hence, there is always
upper bound limit to the usable timestep size for any
unconditionally stable implicit method for aparticular problem
space even though it is still stable beyondthis limit. Here, the
Nyquist criterion relating the Courantnumber, s and mesh density, N
can be given as:
s≤ N2Hence, it can be observed from Fig. 7 that the time step
size
in CNFDTD is more strict and much smaller than Nyquistlimit in
order to achieve desired numerical accuracy withcomparatively less
dispersion errors for a particular problemspace.
VI. CONCLUSION
It can be concluded that with increase in time step size
anddecrease in number of required time steps, there is
significantreduction in computational time in case of
unconditionallystable implicit method viz. ADI-FDTD and CN-FDTD
ascompared to conventional FDTD. However, there is always anupper
bound on maximum time step size in order to maintaindesired level
of numerical accuracy in result for any problemspace.
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