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 WangKey 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]:

� �0 0

1 1 1( )2 � �

� � � �H,E H H + E E (1)

Applying the variation method, Eq.(2) can be rewritten in the following matrix form as:

t�� � �

�A (2)

30 0

30 0

1

1

R

R

� �

� �

� �� �� �� � �� �� �� �

0

�0

, 0

0

ˆ ��

E = E (3a)

978-1-4673-2185-3/12/$31.00 ©2012 IEEE

0z

0

0

y

z x

y x

� �� � � �� �� �� �� �

� � �� �� �� �� � � �� �� �

R = (3b)

Where [ , , ]Tx y zE E E�E and [ , , ]T

x y zH H H�H are

electric and magnetic filed, respectively. ˆ[ , ]T� � H E and

30 is the 3 3 zero matrix and R � is the 3 3 matrix representing the curl operator, 0 0,� � 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:

� � � � � �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 schemeThe 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]:

� � � �1

1

exp exp

exp( )exp( ) ( )m

pl l

l

t t

d t c t O t �

�

� � � �� �� �

� � � � ��

A B C

C B(5)

30

3 3

1 R� � �

� �� � �� �� �

0B

0 0

3 3

30

1 R�

� �� �� � �� �� �

0 0C

0 (6)

Where lc and ld are constant coefficients of the symplectic

integrators. In view of 0� �B and 0� �C ( 2� � ), so the exponential operators exp( )lc t� B and exp( )ld t� Ccan be computed analytical by Taylor series expansion, then Eq.(6) could be rewritten as:

� � 16 6

1

exp ( )( ) ( )m

pl 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 schemeThe matrix operator A was divided into series of real

antisymmetric operators1

s

ii�

� �A A . Then the formulation of

Lie-Trotter-Suzuki approximation can be expressed as[11]:

� �1 1

exp exp( ) lim exp( )nss

ii ni i

tt tn�

� �

�� �� � � � ! "

� �� � AA 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 2

2

22

1

exp ( ) ( )( )2 2

( )( ) ( )2 2

ttt

t O tt

�� � � �

��

� � � � �

AIA A II A

AI II 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) 21

1| [ | | ]M

n n nl s l s l s

sf C f f

# # � �

��

� � � (10)

Where , ,x y z# � and1 2

2 22 2

( 1) [( 1)!!]2 (2 1) ( 1)!( )!

s

s M MM

MCs s s

�

�

�

The coefficients of the fourth order accuracy are 198

C �

and 2124

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

fourth order accuracy scheme will be taken into account.

2) The compact finite differenceThe compact finite difference expressed as[6]:

1/ 2 1/ 21 1 1| | |n n n l l

l l lf ff f f� � $

# # # #�

� �

� � �� � �

� � � � (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 /11� $� � in our numerical experiment.

3) The scaling functionThe 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 ( )x% can be written as[7]:

1 2

, ,

( , , , ) ( 1 2, , ) ( ) ( ) ( ) ( )nx x i j k n

ni j k

E x y z t E i j k x y z h t% % %�

��

� �� (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 nl l s l s

sf a l f f

# #

� �

��

� � � (13)

Where 2 1sL L� ,( 1 2)( ) ( ) xa l x l dx

x%%

�

� � �

�&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 dl1 0.17399689146541 0.623379324513222 -0.12038504121430 -0.123379324513223 0.89277629949778 -0.123379324513224 -0.12038504121430 0.623379324513225 0.89277629949778 0

TABLE II COEFFICIENTS OF THE DAUBECHIES SCALING FUNCTION

D1 D2 D3a(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

1cos( ) 1 [4 ( )]2

mp

p x y zp

t g s' ( ( (�

� � � � �� (14)

Where 1 1 2 2

1 1 1 2 2

1 1 2 21 1 1 2 2

p i j i j ip jpi j i j ip jp m

i j i j ip jpi j i j ip jp m

g c d c d c d

d c d c d c) ) ) ) ) ) ) )

) * * * * * * )

�

�

�

��

�

�

�

The space increment is � ( x y z� � � � � � � ), and the temporal increment is t� , CFL number is s c t� � � .Parameters #( ( , ,x y z# � ) are determined by the spatial difference approximation scheme. Parameters #( have been defined for the high order staggered difference, compact finite difference and Daubechies scaling function, respectively.

1) The high order staggered difference 39 1sin( ) sin( )

8 2 24 2k k# #

#

# #(

� �� (15)

2) The compact finite difference

sin( )2

2 cos( ) 1

k

k

#

##

#$

(� #

�

�� �

(16)

3) The Daubechies scaling function 3 5

(0)sin( ) (1)sin( ) (2)sin( )2 2 2

k k ka a a# # #

#

# # #(

� � �� � � (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 % +� � , sin sinyk k % +� � and coszk k +� � .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

22 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 st

s s s( ( ( ( ( ( ( ( ( (

'( ( (

� � � �� �

� � �

(18 where 2 2 2 2

x y z( ( ( (� � � .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

symplectic and ADI schemes are analyzed firstly as a function of the propagation angle% as shown in Fig.1, in case PPW=10, CFL=0.4 and 3+ ,� . 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 6% ,� and 3+ ,� . 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 %(-)

Rel

ativ

e P

hase

Vel

ocity

Err

or(d

B)

ADI-FDTDADI-MRTDADI-CFDTDS-FDTDS-MRTDS-CFDTD

Fig.1. Numerical dispersion curves as a function of��, for ���/3, PPW=10 and CFL=0.4.

0.20.4

0.60.8

1

5

10

15

20

-120

-100

-80

-60

-40

-20

CFL(c�t/�x)PPW(./�x)

Rel

ativ

e P

hase

Vel

ocity

Err

or(d

B)

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 �=�/6�and�����/3.

REFERENCES

[1] K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media,” IEEE Trans. on Antennas and Propagation, vol. 14, pp. 5, 1966.

[2] N. Anderson, and A. M. Arthurs, “Helicity and variational principles for Maxwell's equations,” International Journal of Electronics, vol. 54, no. 6, pp. 861-864, 1983/12/01, 1983.

[3] T. Namiki, “3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell's equations,” Microwave Theory and Techniques, IEEE Transactions on, vol. 48, no. 10, pp. 1743-1748, 2000.

[4] L. Kang, L. Yaowu, and L. Weigan, “A higher order (2,4) scheme for reducing dispersion in FDTD algorithm,” Electromagnetic Compatibility, IEEE Transactions on, vol. 41, no. 2, pp. 160-165, 1999.

[5] N. V. Kantartzis, and T. D. Tsiboukis, "Higher-Order FDTD Schemes for Waveguide and Antenna Structures," San Rafael, CA,USA:Morgan & Claypool Publishers, 2006.

[6] J. L. Young, D. Gaitonde, and J. J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 11, pp. 1573-1580, Nov, 1997.

[7] M. Krumpholz, and L. P. B. Katehi, “MRTD: New time-domain schemes based on multiresolution analysis,” IEEE Transactions on Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-571, Apr, 1996.

[8] W. E. I. Sha, Wu Xianliang, Huang Zhixiang, and Chen Mingsheng, TheHigh-Order Symplectic Finite-Difference Time-Domain Scheme: Passive Microwave Components and Antennas, Vitaliy Zhurbenko (Ed.),INTECH, 2010.

[9] H. Yoshida, “Construction of higher order symplectic integrators,” Physics Letters A, vol. 150, no. 5-7, pp. 262-268, 1990.

[10] W. E. I. Sha, Z. X. Huang, M. S. Chen et al., “Survey on symplectic finite-difference time-domain schemes for Maxwell's equations,” IEEE Transactions on Antennas and Propagation, vol. 56, no. 2, pp. 493-500, 2008.

[11] X. L. Wu, and Z. X. Huang, “Research on the unified treatment of time-domain electromagnetic simulation,” Sciencepaper Online, vol. 5, no. 1, pp. 62-67, 2010.

[12] M. Darms, R. Schuhmann, H. Spachmann et al., “Dispersion and asymmetry effects of ADI-FDTD,” IEEE Microwave and Wireless Components Letters, vol. 12, no. 12, pp. 491-493, Dec, 2002.

[13] K. L. Shlager, and J. B. Schneider, “Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes,” IEEE Transactions on Antennas and Propagation, vol. 52, no. 4, pp. 1095-1104, Apr, 2004.

[14] X. G. Ren, Z. X. Huang, X. L. Wu, S. L. Lu, H. Wang, L. Wu, and S. Li, "High-order unified symplectic FDTD scheme for the metamaterials," Computer Physics Communications, vol. 183, pp. 1192-1200, 2012.