Modified Splitting FDTD Methods for Two-Dimensional Maxwell ’ s Equations

In this paper, we develop a new method to reduce the error in the splitting finite-difference method of Maxwell’s equations. By this method two modified splitting FDTD methods (MS-FDTDI, MS-FDTDII) for the two-dimensional Maxwell equations are proposed. It is shown that the two methods are second-order accurate in time and space and unconditionally stable by Fourier methods. By energy method, it is proved that MS-FDTDI is second-order convergent. By deriving the numerical dispersion (ND) relations, we prove rigorously that MS-FDTDI has less ND errors than the ADI-FDTD method and the ND errors of ADI-FDTD are less than those of MS-FDTDII. Numerical experiments for computing ND errors and simulating a wave guide problem and a scattering problem are carried out and the efficiency of the MS-FDTDI and MS-FDTDII methods is confirmed.


Introduction
The finite-difference time-domain (FDTD) method for Maxwell's equations, which was first proposed by Yee (see [1], also called Yee's scheme) in 1966, is a very efficient numerical algorithm in computational electromagnetism (see [2]) and has been applied in a broad range of practical problems by combining absorbing boundary conditions (see [3][4][5][6][7] and the references therein).It is well known from [8] that the Yee Scheme is stable when time and spatial step sizes (Δ, Δ, and Δ for 2D case) satisfy the Courant-Friedrichs-Lewy (CFL) condition Δ ≤ [1/(Δ) 2 + 1/(Δ) 2 ] −1/2 , where  is the wave velocity.To overcome the restriction of the CFL condition there are many research works on this topic; for example, see [9][10][11][12][13][14][15][16][17] and the references therein.In [15], two unconditionally stable FDTD methods (named as S-FDTDI and S-FDTDII) were proposed by using splitting of the Maxwell equations and reducing of the perturbation error, where S-FDTDII, based on S-FDTDI (first-order accurate), is second-order accurate and has less numerical dispersion (ND) error than S-FDTDI.However, the second convergence of S-FDTDII was not proved by the energy method.
In this letter, by introducing a new method to reduce the error caused by splitting of equations [15] (other methods of reducing perturbation error caused by splitting of differential equations can be seen in [18]), we propose two modified splitting FDTD methods (called MS-FDTDI and MS-FDTDII) for the 2D Maxwell equations.It is proved by the energy method that MS-FDTDI with the perfectly electric conducting boundary conditions is second-order convergent in both time and space.By Fourier method we derive the amplification factors and ND relations of MS-FDTDI and MS-FDTDII.Then, we prove that these two methods are unconditionally stable and that MS-FDTDI has less ND errors than S-FDTDII (or ADI-FDTD [10,11]).Numerical experiments to compute numerical dispersion errors and convergence orders and to simulate a scattering problem are carried out.Computational results confirm the analysis of MS-FDTDI and MS-FDTDII.

Remark 1.
(1) In order to see the difference between MS-FDTDTI and the S-FDTDII method in [15], we give the equivalent forms of the two methods: where ( 11)-( 13) with  = 0 being the equivalent form of S-FDTDII (Stage 1 of S-FDTDII is the same as (6); Stage 2 of S-FDTDII is ( 7)-( 8) with the last term on the right hand side of (7) removed); ( 11)-( 13) with the case is the equivalent form of MS-FDTDI.By these forms we see that MS-FDTDI is different from the S-FDTDII and ADI-FDTD methods (see [10,11], where splitting of the equations is not used; however, the equivalent form of S-FDTDII is the same as that of 2D ADI-FDTD).
(2) MS-FDTDI has similar perturbation term as the D'yakonov scheme (see [19]).The equivalent form of this scheme is (11)- (13) with and the perturbation term on the right hand side of (12) being removed.In the comparison of these equivalent forms we see that the perturbation term and its location of MS-FDTDI are different from those of the D'yakonov's scheme.This implies that they are different.
The boundary and initial conditions of MS-FDTDII are the same as MS-FDTDI.
The equivalent form of MS-FDTDII is ( 11)-( 13) with By these equivalent forms we see that MS-FDTDI and MS-FDTDII are of second-order accuracy.

Analysis of Stability and Numerical Dispersion Error
In this section we first derive the amplification factors and numerical dispersion (ND) relations of MS-FDTDI and MS-FDTDII and then we analyze the stability and ND error.

Stability Analysis.
Let the trial time-harmonic solution of the Maxwell equations be where   = √ −1 is the unit of complex numbers,  1 ,  2 , and  0 are the amplitudes,   and   are the wave numbers along the -axis and -axis, and  is the amplification factor.Substituting the above expressions into the equivalent form of MS-FDTDI and evaluating the determinant of the coefficient matrix of the resulting system of equations for  1 ,  2 , and  0 , we get a quadratic equation of .Solving this equation yields the amplification factors for MS-FDTDI: where the coefficients are The modulus of  1 or  2 is equal to one, implying that MS-FDTDI is unconditionally stable and nondissipative.
Similarly, we obtain the amplification factors of MS-FDTDII: where  0 is the same as that in (20), and d1 is implies that MS-FDTDII is also unconditionally stable and nondissipative.Remark 3. The amplification factors of S-FDTDII, which are the same as those of ADI-FDTD (the derivation is seen in [15]), are where  0 is the same as that in (20), and  1 is

Numerical Dispersion Analysis.
Let  = 1/ √  be the wave speed.Substituting  =  Δ into (20), we obtain the ND relation of MS-FDTDI: where   and   are defined under (20).
Similarly, the ND relation of MS-FDTDII is Remark 4. The ND relation of S-FDTDII is the same as that of ADI-FDTD (see [15]), which is By using the Taylor expansions of sin() and cos() and the continuous dispersion relation: By the second and third terms of truncation errors we see that   <   , implying that the ND error of S-FDTDII or ADI-FDTD is less than that of MS-FDTDII.Noting that we obtain that the ND error of MS-FDTDI is less than that of S-FDTDII (or ADI-FDTD).6)- (8).

Error Estimates and Convergence of MS-FDTDI
Subtracting the equivalent form of MS-FDTDI ( 11)-( 13) from the discretized Maxwell equations (whose form is like ( 11)-( 13) with extra truncation errors), we obtain the following error equations: where for  = ,  and  = 0,  (2) By the similar method to the above it can not be proved that MS-FDTDII is convergent since the perturbation terms in this scheme are not controlled.

Numerical Experiments
We do some experiments to compute the ND errors of MS-FDTDI and MS-FDTDII, to solve a wave guide problem, and to simulate a scattering problem by the two methods.

Computation of Numerical Dispersion Errors.
Let  be the wave length, Δ = Δ = ℎ, and   = /ℎ be the number of points per wavelength, and  = (Δ)/ℎ is a multiple of the CFL number (CFL number equals √ 2Δ/ℎ in this case);  is the wave propagation angle.Then, by   =  cos(),   =  sin(),  2 =  2  +  2  ,  = 2/, and the expressions of   and   (defined in Section 3.1), we see that the amplification or stability factor  is a function of , , and   ; that is,  = (, ,   ).The ND errors of MS-FDTDI, MS-FDTDII, and S-FTTDII are computed by the following formula (see [20]): where I() and R() denote the imaginary and real parts of the amplification factor .We plot the normalized phase velocity V  / with respect to , , and   (see Figures 1-2).
Figure 1 shows the variation of V  / against the wave propagation  with   = 40 and  = 3.5 for MS-FDTDI, MS-FDTDII, and S-FDTDII.From the curves we see that V  / for MS-FDTDI is more close to 1 than that of S-FDTDII (or ADI-FDTD; S-FDTDII is equivalent to ADI-FDTD), and the latter is more close to 1 than MS-FDTDII.This means that the ND error of MS-FDTDI is less than that of S-FDTDII (or ADI-FDTD) and that the ND error of S-FDTDII (or ADI-FDTD) is less than that of MS-FDTDII.
Figures 2 and 3 give the graphs of V  / against   with  = 35 ∘ and  = 1.5 and against  with  = 35 ∘ and   = 60, respectively.From these curves we see the same conclusion as that drawn from Figure 1.
The experimental results are shown in Tables 1-2, where the drive routines are written in Fortran, and the computation was run on a 2.53 GHz PC having 2.0 GB RAM and Windows 7 operating system.
Table 1 gives the relative errors Re-Err-E and Re-Err-H and convergence orders of the approximate electric and magnetic fields computed by MS-FDTDI, MS-FDTDII, and S-FDTDII at time  = 1 with Δ = Δ = Δ = ℎ = 0.02, 0.01, 0.005 (the results computed by ADI-FDTD are the same as S-FDTDII).By the relative errors we see that MS-FDTDI is more accurate than S-FDTDII (or ADI-FDTD) and that S-FDTDII (or ADI-FDTD) is more accurate than MS-FDTDII.The computed convergence orders of these three methods are approximately equal to 2, implying that MS-FDTDI and MS-FDTDII are second methods.This is consistent with the analysis in theory.
To see the long time behavior, Table 2 lists the relative errors and CPU time (1/ seconds) in the cases Δ = Δ = Δ = 0.01 and different time lengths  = 10, 20, 40.From this table we see that MS-FDTDI is better than S-FDTDI, that S-FDTDII is better than MS-FDTDII in a long time computation, and that the CPU time for the three methods is of a little difference.Let  be the distance from a point to (0, 0.5).The point source used to be the initial fields is defined by
To compute this problem we use the perfectly matched sponge layers (see [7]) to be placed in the upper (2 ≤  ≤ 2.2) and lower (−0.2 ≤  ≤ 0) parts of the open domain with electric and magnetic losses (see [20]): and the Maxwell equations in the sponge layers [7] are The implementation of MS-FDTDI for (42) in the sponge layer is given in the following two stages.

𝐸 𝑛+1
− The implementation of S-FDTDII and Yee Scheme in the sponge layer is seen in [15].We take the step sizes Δ = Δ = Δ = 0.01 and do the computation by MS-FDTDI, MS-FDTDII, S-FDTDII, and Yee Scheme.The contours of the numerical magnetic fields    are plotted in Figures 4-5.
Figures 4 and 5 give the contours of    with   = 0.2, 0.5 and 1 obtained by MS-FDTDI, MS-FDTDII, S-FDTDII, and Yee's scheme.Comparing these figures, we find that the numerical solutions computed by MS-FDTDI and MS-FDTDII are in agreement with those by S-FDTDII and Yee's scheme.
Also clearly shown in the contours at  = 0.2 and 0.5 of Figures 4-5 are the reflected and transmitted wavefronts at the dielectric interface and the absorption in the PMLs layer in the bottom of the rectangle for the four methods.The symmetry of the curves on the left and right hand sides reflects the use of periodic boundary conditions.This confirms that MS-FDTDI and MS-FDTDII are effective in solving the scattering problem.

Conclusions and Remarks
In this letter we proposed two FDTD methods (MS-FDTDI and MS-FDTDII) for the 2D Maxwell's equations by introducing two new methods to reduce the perturbation error caused by splitting of Maxwell equations.It was shown that the two methods are second-order accurate.By energy method MS-FDTDI was proved to be second-order convergent.By Fourier method the unconditional stability of the two methods was proved and the numerical dispersion (ND) relations were derived.By analyzing the truncation errors of the ND relations, we proved rigorously that MS-FDTDI has less ND error than S-FDTDII or ADI-FDTD and that the ND error of S-FDTDII or ADI-FDTD is less than that of MS-FDTDII.Numerical experiments were carried out, and the analysis on stability and ND error as well as the efficiency of MS-FDTDI and MS-FDTDII in computing a wave guide problem and a scattering problem was confirmed.The new methods for reducing splitting errors and analyzing ND errors could be used for construction of other methods and ND analysis.
velocity against N  Number of points per wavelength N
Contour of Hz by MS-FDTDI scheme by t = 0.2 Contour of Hz by MS-FDTDI scheme by t = 0.5 Contour of Hz by MS-FDTDI scheme by t = 1Contour of Hz by MS-FDTDII scheme by t = 0.2 Contour of Hz by MS-FDTDII scheme by t = 0.5 Contour of Hz by MS-FDTDII scheme by t =
we derive the main truncation errors of the ND relations of MS-FDTDI, MS-FDTDII, and S-FDTDII, denoted by   ,   , and   , which are
The MS-FDTDII scheme for the Maxwell equations (42) is given in the following two stages.Stage 1. Stage 1 is the same as Stage 1 of MS-FDTDI.
where the subscripts of the fields for the spatial indexes are omitted for the simplicity in notation.