This document discusses numerical dispersion analysis of symplectic and alternating direction implicit (ADI) schemes for computational electromagnetic simulation. It presents Maxwell's equations as a Hamiltonian system that can be written as symplectic or ADI schemes by approximating the time evolution operator. Three high order spatial difference approximations - high order staggered difference, compact finite difference, and scaling function approximations - are analyzed to reduce numerical dispersion when combined with the symplectic and ADI schemes. The document derives unified dispersion relationships for the symplectic and ADI schemes with different spatial difference approximations, which can be used as a reference for simulating large scale electromagnetic problems.

1. Numerical Dispersion Analysis of Symplectic and
ADI Schemes
Xin-gang Ren#
, Zhi-xiang Huang*
, Xian-liang Wu,
Si-long Lu, Yi-cai Mei, Hong-mei Du, Hui Wang
Key Lab of Intelligent Computing & Signal Processing
Ministry of Education, Anhui University
Hefei, China
#xingangahu@126.com *zxhuang@ahu.edu.cn
Xian-liang Wu, Jing Shen
Department of Physics and Electronic Engineering
Hefei Normal University
Hefei, China
xlwu@ahu.edu.cn
Abstract-In this paper, Maxwell’s equations are taken as a
Hamiltonian system and then written as Hamiltonian canonical
equations by using the functional variation method. The
symplectic and ADI schemes, which can be extracted by applying
two types of approximation to the time evolution operator, are
explicit and implicit scheme in computational electromagnetic
simulation, respectively. Since Finite-difference time-domain
(FDTD) encounter low accuracy and high dispersion, the more
accurate simulation methods can be derived by evaluating the
curl operator in the spatial direction with kinds of high order
approaches including high order staggered difference, compact
finite difference and scaling function approximations. The
numerical dispersion of the symplectic and ADI schemes
combining with the three high order spatial difference
approximations have been analyzed. It has been shown that
symplectic scheme combining with compact finite difference and
ADI scheme combining with scaling function performance better
than other methods. Both schemes can be usefully employed for
simulating and solving the large scale electromagnetic problems.
I. INTRODUCTION
The finite-difference time domain (FDTD) method, which
was firstly proposed by K.S.Yee[1]and has widely been used
for solving the electromagnetic problems. The central
difference was used to approximate the temporal and spatial
derivate in time domain Maxwell’s equations. Since the
FDTD method is based on the explicit difference, the Courant
stability condition must be satisfied in order to guarantee
numerical stability. The time step size must be very small in
order to obtain a high accuracy when a large scale size
structure was simulated. The numerical dispersion error will
be accumulated with an increased simulation time step. The
high order FDTD scheme is proposed to reduce the dispersion
but will encounter a low stability. Researchers have carried
out many improvements to overcome the shortcoming. The
one is the symplectic scheme which have been proved to
enhance the stability and reduce the dispersion error because
of energy conservation of Hamiltonian system[2]. Besides the
alternating-direction implicit (ADI) scheme which is
unconditionally stable is proposed to manipulate the large
scale size problems and theoretically there is no limitation on
the time step size [3].But as the size of time step is increased,
numerical dispersion errors will become large. To overcome
this shortcoming, the more accurate curl operator
approximations have been employed to result in a low
numerical dispersion. Three types of approximations are
considered in the following context. The high order staggered
difference which is also an explicit scheme can obtain a better
dispersion than the central difference [4, 5]. The compact
finite difference which is an implicit scheme approximate will
result in the solution of a tri-diagonal matrix and lead to a low
numerical dispersion [6]. The multiresoultion time domain
(MRTD) method which is based on the scaling function has
been broadly accepted as a high accurate method to improve
the numerical dispersion [7].
In this paper, Maxwell’s equations firstly are written as
Hamiltonian canonical equations by using the functional
variation method. The symplectic and ADI schemes have been
extracted by applying two types of approximation to the time
evolution operator. The high order staggered difference,
compact finite difference and scaling function approximations
are taken to approximate the spatial curl operators to obtain
low numerical dispersion errors. The unified dispersion
relationships are derived for the symplectic and ADI schemes,
respectively. The numerical dispersion is studied by applying
different curl operator approximation. The result can be used
as a reference when simulate and solve the large scale
electromagnetic problems.
II. FORMULATION
Maxwell’s equations can be rewritten in the form of
Hamilton function as[2, 8]:

2. 0 0
1 1 1
( )
2 P H
+ ˜’u ˜’uH,E H H + E E (1)
Applying the variation method, Eq.(2) can be rewritten in
the following matrix form as:
t
w
) )
w
A (2)
3
0 0
3
0 0
1
1
R
R
H P
H P
§ ·
¨ ¸
¨ ¸
¨ ¸
¨ ¸¨ ¸
© ¹
0
ǹ
0
, 0
0
ˆ H
P
E = E (3a)
978-1-4673-2185-3/12/$31.00 ©2012 IEEE

3. 0
z
0
0
y
z x
y x
w w§ ·
¨ ¸w w
¨ ¸
w w¨ ¸
’u¨ ¸w w
¨ ¸
w w¨ ¸¨ ¸w w© ¹
R = (3b)
Where [ , , ]T
x y zE E EE and [ , , ]T
x y zH H HH are
electric and magnetic filed, respectively. ˆ[ , ]T
) H E and
30 is the 3 3u zero matrix and R ’u is the 3 3u matrix
representing the curl operator, 0 0,H P are the permittivity
and permeability in the vacuum, respectively.
A. The time evolution matrix
From Eq.(3), the time evolution of the electromagnetic
field from 0t to t t' can be accurately obtained by the
exponential operator exp( )t'A as following:

6. exp 0t t) ' ' )A (4)
However, the exponential operator exp( )t'A cannot be
evaluated at any t' . Fortunately, there are mainly two
approximations will deduce to lots of simulation methods
which have been widely used, one is the use of symplectic
propagator technique which will extract the explicit
symplectic scheme and the other is the use of
Lie-Trotter-Suzuki approximation which will extract the
implicit ADI scheme.
1) Symplectic scheme
The operator exp( )t'A is approximated with the
symplectic propagator technique by splitting matrix A into
two noncommuting operators ,B C , i.e. A B C and , then
a m-stage and p-order approximation can be obtained in the
following product form of the exponential operator[9,14]:

8. 1
1
exp exp
exp( )exp( ) ( )
m
p
l l
l
t t
d t c t O t
' 'ª º¬ ¼
' ' '–
A B C
C B
(5)
3
0
3 3
1
R
P
§ ·
¨ ¸
¨ ¸
¨ ¸
© ¹
0
B
0 0
3 3
3
0
1
R
H
§ ·
¨ ¸
¨ ¸
¨ ¸
© ¹
0 0
C
0
(6)
Where lc and ld are constant coefficients of the symplectic
integrators. In view of 0D
B and 0D
C ( 2D t ), so
the exponential operators exp( )lc t' B and exp( )ld t' C
can be computed analytical by Taylor series expansion, then
Eq.(6) could be rewritten as:

9. 1
6 6
1
exp ( )( ) ( )
m
p
l l
l
t d t c t O t
' ' ' '–A I B I C
(7)
The value of the symplectic integrator coefficients can
be found in Ref[10]. Especially, one can find that the
symplectic scheme can be reduced to the conventional
FDTD method when the symplectic integrator coefficients
are chosen as 1 2 1/ 2c c ; 1 1d , 2 0d . Here we use
the coefficient as shown in Table.I, then a fourth order
accuracy will be gained in the temporal differential
approximation.
2) ADI scheme
The matrix operator A was divided into series of real
antisymmetric operators
1
s
i
i
¦A A . Then the formulation of
Lie-Trotter-Suzuki approximation can be expressed as[11]:

10. 1 1
exp exp( ) lim exp( )
nss
i
i
n
i i
t
t t
nof
'ª º
' ' « »
¬ ¼
¦ –
A
A A (8)
Especially, if we set the parameters 2, 2s n and apply
the Pade approximation, Eq.(9) will be reduced to a simple
form[12]:

11. 1
1 2
2
22
1
exp ( ) ( )( )
2 2
( )( ) ( )
2 2
t
t
t
t
O t
t
'
'
'
'
˜ '
'
AI
A A I
I A
AI
I
I A
(9)
It can be proved that Eq.(10) is the time evolution operator
of the implicit and unconditional stable ADI scheme.
B. The Spatial difference approximation
There are kinds of methods to approximate the spatial
derivate, but three types of high accurate method will be
considered in the following including high order staggered
difference, compact finite difference and scaling function
approximations. Firstly, , ,| ( , , ; )n
i j kf f i x j y k z n t' ' ' '
was denoted to approximate the exact solution ( , , )f x y z at
point ( , , )i x j y k z' ' ' in the n-th time step.
1) The high order staggered difference
The high order accuracy discretized scheme can be express
as[11]:
/2
(2 1) 2 (2 1) 2
1
1
| [ | | ]
M
n n n
l s l s l s
s
f C f f
[ [
w
w '
¦ (10)
Where , ,x y z[ and
1 2
2 2
2 2
( 1) [( 1)!!]
2 (2 1) ( 1)!( )!
s
s M MM
M
C
s s s
The coefficients of the fourth order accuracy are 1
9
8
C
and 2
1
24
C . A low dispersion error will be achieved by
applying the high order staggered difference so it can be done
with the large scale problem, while the low
Courant–Friedrichs–Levy (CFL) number is the drawback. The

12. fourth order accuracy scheme will be taken into account.
2) The compact finite difference
The compact finite difference expressed as[6]:
1/2 1/2
1 1 1| | |n n n l l
l l l
f ff f f
D D E
[ [ [ [
w w w
w w w '
(11)
where , ,x y z[ , and a fourth order accuracy of the spatial
difference can be given by setting the compact finite
difference coefficients 1/ 22, 12 /11D E in our numerical
experiment.
3) The scaling function
The multiresolution time domain (MRTD) which based on
Daubechies scaling functions is proposed to enhance stability
and reduce the numerical dispersion. The mainly idea is that
electromagnetic field component, taking xE for example,
expansion with the Daubechies compact support scaling
function ( )xI can be written as[7]:
1 2
, ,
( , , , ) ( 1 2, , ) ( ) ( ) ( ) ( )n
x x i j k n
n
i j k
E x y z t E i j k x y z h tI I I
f
f
¦ (12)
where ( ) ( 1 2)nh t h t t n' , ( )h t is the Haar wavelet scaling
function. The other field components can be obtained with a
similar way. Substituting expression of the field components
into the Maxwell equation with the application of the Galerkin
method and the vanishing moment L , then the spatial
difference can be expressed in a similar way of the high order
staggered difference as:
1
0
1
| ( )[ | | ]
sL
n n n
l l s l s
s
f a l f f
[ [
w
w '
¦ (13)
Where 2 1sL L ,
( 1 2)
( ) ( )
x
a l x l dx
x
I
I
f
f
w
w³
The coefficients of the Daubechies compact support scaling
function are listed in Table .II. [13], in case of 0,2,4s
corresponding to 1 2 3, ,D D D . Here, 2D will be used to
analyze the numerical dispersion.
TABLE I
COEFFICIENTS OF THE SYMPLECTIC INTEGATOR PROPAGATORS
cl dl
1 0.17399689146541 0.62337932451322
2 -0.12038504121430 -0.12337932451322
3 0.89277629949778 -0.12337932451322
4 -0.12038504121430 0.62337932451322
5 0.89277629949778 0
TABLE II
COEFFICIENTS OF THE DAUBECHIES SCALING FUNCTION
D1 D2 D3
a(0) 1 1.229166667 1.291812928
a(1) -0.093750000 -0.137134347
a(2) 0.010416667 0.028761772
a(3) -0.003470141
a(4) 0.000008027
C. The numerical dispersion relationship
The phase velocity of the simulation wave will slightly
differ from the phase velocity of the natural media when the
electromagnetic problem is simulated by a numerical method.
The phase velocity will be varied with the frequency, direction
of propagation, spatial and temporal increment. The numerical
dispersion of the symplectic and ADI schemes were briefly
given in following.
The numerical dispersion relationship of the symplectic
scheme can be expressed as[10]:
2 2 2 2
1
1
cos( ) 1 [4 ( )]
2
m
p
p x y z
p
t g sZ K K K' ¦ (14)
Where
1 1 2 2
1 1 1 2 2
1 1 2 2
1 1 1 2 2
p i j i j ip jp
i j i j ip jp m
i j i j ip jp
i j i j ip jp m
g c d c d c d
d c d c d c
d d d d d d d d
d d
¦
¦
The space increment is ' ( x y z' ' ' ' ), and the
temporal increment is t' , CFL number is s c t' ' .
Parameters [K ( , ,x y z[ ) are determined by the spatial
difference approximation scheme. Parameters [K have been
defined for the high order staggered difference, compact finite
difference and Daubechies scaling function, respectively.
1) The high order staggered difference
39 1
sin( ) sin( )
8 2 24 2
k k[ [
[
[ [
K
' '
(15)
2) The compact finite difference
sin( )
2
2 cos( ) 1
k
k
[
[
[
[
E
K
D [
'
'
(16)
3) The Daubechies scaling function
3 5
(0)sin( ) (1)sin( ) (2)sin( )
2 2 2
k k k
a a a
[ [ [
[
[ [ [
K
' ' '
(17)
If k represents the numerical wave-number, then the
numerical wave-number in , ,x y z direction can be defined
as cos sinxk k I T , sin sinyk k I T and coszk k T .
The numerical dispersion formula of the ADI scheme can
be given by the relation:
2 2 2 2 2 2 2 2 2 2 2 2 2
2
2 2 2 2 2 2 2
4 [ ( ][1 ]
sin ( )
[(1 )(1 )(1 )]
x y y z z x x y z
x y z
s s s
t
s s s
K K K K K K K K K K
Z
K K K
'
(18
where 2 2 2 2
x y zK K K K .
The numerical dispersion of the symplectic finite difference
time domain (S-FDTD), symplectic compact finite difference
time domain (S-CFDTD) and symplectic multiresolution time
domain (S-MRTD) can be obtained by substituting (15), (16),
(17) into (14), respectively. The numerical dispersion of ADI
finite difference time domain (ADI-FDTD), ADI compact
finite difference time domain (ADI-CFDTD) and ADI
multiresolution time domain (ADI-MRTD) were obtained by
substituting(15), (16), (17) into (18), respectively.
III. NUMERICAL VALIDATION
The relative phase velocity error of the aforementioned

13. symplectic and ADI schemes are analyzed firstly as a
function of the propagation angleI as shown in Fig.1, in case
PPW=10, CFL=0.4 and 3T S . Then for a better
understanding of the dispersion, the relative phase velocity
error was taken as a function of PPW and CFL number with
a fixed propagation angle 6I S and 3T S . The
results reveal that the S-CFDTD scheme has the lowest
numerical dispersion curve, and the dispersion curve of
ADI-MRTD scheme is better than other ADI schemes
especially at a low PPW number and small propagation angle.
That means both ADI-MRTD and S-CFDTD have a high
computational precision and can be used to simulate the
large scale size electromagnetic problems.
IV. CONCLUSION
In this paper, Maxwell’s equations are taken as a
Hamiltonian system and then written as Hamiltonian
canonical equations by using the functional variation method.
The symplectic and ADI schemes, which can be extracted by
applying two types of approximations to the time evolution
operator, are explicit and implicit scheme in computational
electromagnetic simulation, respectively. Then the unified
dispersion relationships are derived for the symplectic and
ADI scheme, respectively. The numerical dispersion is studied
by applying three types of high order spatial difference
approximations. It has been shown in the dispersion curves
that symplectic scheme combining with compact finite
difference and ADI scheme combining with scaling function
performance a better dispersion than other methods. Both
schemes can be usefully employed for simulating and solving
the large scale electromagnetic problems.
ACKNOWLEDGMENT
The authors gratefully acknowledge the support of the NSFC
of China (60931002, 61101064), Distinguished Natural
Science Foundation (1108085J01), and Universities Natural
Science Foundation of Anhui Province (No. KJ2011A002,
KJ2011A242), and Financed by the 211 Project of Anhui
University.
0 10 20 30 40 50 60 70 80 90
-90
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
Propagation Angle I(q)
RelativePhaseVelocityError(dB)
ADI-FDTD
ADI-MRTD
ADI-CFDTD
S-FDTD
S-MRTD
S-CFDTD
Fig.1. Numerical dispersion curves as a function of I, for T S/3, PPW=10
and CFL=0.4.
0.2
0.4
0.6
0.8
1
5
10
15
20
-120
-100
-80
-60
-40
-20
CFL(c't/'x)PPW(O/'x)
RelativePhaseVelocityError(dB)
S-CFDTD
ADI-FDTD
S-MRTTD
ADI-MRTD
ADI-CFDTD
S-FDTD
Fig.2. Numerical dispersion curves as a function of PPW and CFL, for
I=S/6 and T S/3.
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