The Stability and Numerical Dispersion Study of the ADI-SFDTD Algorithm

In this letter, the alternating-direction-implicit (ADI) technique is applied to Symplectic finite-difference time-domain (SFDTD) method, the curl operator is endued with two different styles when doing computation from the ( 1) s  th progression to s th progression. It holds the advantages of both ADI-FDTD and SFDTD, not only eliminating the restriction of the Courant-Friedrich-Levy (CFL), but also holding the inner characteristics of Maxwell’s equations. The analytical accuracy and efficiency of the proposed method is verified good.


Introduction
FDTD method is a very useful numerical simulation technique for solving electromagnetic questions.As we know, the traditional FDTD method is based on the explicit finite-difference algorithm, hence, it is limited by Courant-Friedrich-Levy (CFL) stability condition.In order to eliminate the Courant-Friedrich-Levy (CFL) condition restraint, Unconditionally sTable algorithm ADL-FDTD( the alternating-direction-implicit technique finite difference time domain) has been proposed.But in the common ADI-FDTD method, the choice of large time intervals leads to substantial dispersion errors that degrade its performance (A.P. Zhao, 2000;F. Zheng et al., 2000;Huang Z X et al., 2007).
Effective studies (M Kusaf et al., 2005;Ruth F D, 1983) revealed that Maxwell's equations can be viewed as an infinite dimension Hamilton system.FDTD and ADI-FDTD destroy Maxwell's equations' Symplectic structure, so they are not good algorithms for Maxwell's equations' numerical simulation.A good algorithm must hold Maxwell's equations' Symplectic structure.
In this paper a novel algorithm that bases on SFDTD (symplectic finite difference time domain) and ADI has been proposed.We transform Maxwell's equations to Hamilton's equations, and use symplectic propagation technique disperse Hamilton's equations in time domain, and use the ADI technique to discredited Hamilton's equations' curl operator R in spatial domain, then, we discuss the ADI-SFDTD algorithm's stability and numerical dispersion systemically, finally we validate the proposed ADI-SFDTD formulation by a numerical example.

Hamilton transform of Maxwell's equations
In a linear, homogeneous, and isotropic medium, Maxwell's equations can be written as (J.W. Thomas, 1995): Where,  is medium's permittivity and  is medium's permeability.In the Hamilton system, Maxwell's equations can be written as Where, Hamilton function ( , ) H B D is defined as The In the equation (5),  is the inspection function.

ADI-SFDTD method
Studies revealed that Hamilton's equations can be transformed into (6), written as Where,

R
In time domain, from time 0 t  to time t   , the results of ( 6) can be written as For exponential operator can't be used to compute, the exponential operator is approximate to (8) by using symplectic propagation technique. Where, , ( ) m p m p  are symplectic propagation's progression and order.According to (J.W. Thomas, 1995), choose the suiTable propagation sub coefficients { } s c and{ } s d , it can preserve Maxwell's equations' inner characters.In this paper, we use the optimized 5 progression and 4 order propagation sub coefficients.
For ( ) 0,( , , ) , the exponential operators exp( ) and exp( ) L and B L , so in order to get the numerical results of Maxwell's equations, we must discredited the equations in spatial domain again.
Introducing the plane wave' propagation equation, written as: 0 ( , , , ) exp( ( )) In the spherical coordinate system: 0 sin cos The positive direction of  is that of the right-handed rotation from x to y about z axis, the positive direction of  is from the positive z axis towards the negative z axis, closed-form solution at discretization point ( , , ) i x j y k z    in the n th time step.There, Every time step need m progression to simulate and the time increment of the s th progression to Applying the ADI principle into R , we define two different curl operators about R , marked as 1 R and 2 R in following.In ( 1,2)     

R
, the I indicates implicit form and E indicates explicit form.
At s th progression of n th time step, in x-direction, the implicit form is defined in equation ( 12a) and the explicit form is defined in (12b).
, , ( ) In the same way, y-direction's implicit form is , , and explicit form is , , and explicit form is , , . Substituting (10) into (12a) and (12b), we can get 0 0 , , Where, exp( ) 2 In the same way we can get Substituting ( 13a) and ( 13b) into (11a) and (11b), we get 1 2 ( 1) 15) Equation ( 7) can be written as as the fist step and as the second step. For as the fist step and As the second step, where 4 ( 1) .For other components, the equations can be obtained in the same way.

Stability analysis
According to ( 16), growth matrix G can be presented as the product of the first procedure growth matrix 1 G and the second procedure 2 G , written as:  indicates the growth factor of the total procedure, 1  indicates growth factor of the first procedure and 2  indicates growth factor of the second procedure.According to the principle of the matrix growth, we obtain that 1  satisfies equation (21a) and 2  satisfies equation (21b): By solving (21a) and (21b), the growth factor of the first procedure 1 Finally, 1  and 2  yields  , which indicates the growth factor of the total procedure as follows: Equation ( 24) is always satisfied, so that the ADI-STDTD is unconditionally sTable in any case.

Numerical dispersion
Now we assume 1 2 s      of (21a) and (21b) [9].By adding (21a) and (21b) we then get This suggests that numerical dispersion of the ADI-SFDTD method can be reduced to any degree if appropriate cells are used.Figure 1 is the normalized phase velocity of different  for different FDTD schemes, we see the proposed ADI-FDTD has good performance.Figure 2 shows the normalized error contrast between ADI-FDTD and ADI-SFDTD, it clearly shows that ADI-SFDTD is more efficiency.

Conclusion
In this paper , a novel algorithm that based on SFDTD and ADI technique has be proposed, the spatial discretization scheme curl operator R is endued with two different styles when doing computation from the ( 1) s  th progression to s th progression.Then, Its stability and numerical dispersion has been analyzed.The results show that the proposed method has good efficiency and accuracy.
Figure 2. The normalized numerical dispersion error contrast between ADI-FDTD and ADI-SFDTD