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One-step leapfrog ADI-FDTD method for lossy media and its stability analysis.

1. INTRODUCTION

Finite-difference time-domain (FDTD) method has been widely developed to simulate various EM problems in the past decades [1]. However, it is constrained by the Courant-Friedrich-Levy (CFL) stability condition. To overcome this problem, the unconditionally sable alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method was developed [2, 3]. But it employs a split time-step scheme where mid time-step computations are required. As a result, the required memory and CPU time are more than those of the conventional FDTD method. Recently, one-step leapfrog ADI-FDTD method was developed from the conventional ADI-FDTD method [4] where no mid time-step computations are needed. Therefore, it has better computational efficiency [5, 6].

On the other hand, both the FDTD and ADI-FDTD methods are widely studied for lossy media [7-10]. In [11], the leapfrog ADI-FDTD method was also developed for lossy media. But the difference equations are derived by considering the leapfrog ADI-FDTD method as a perturbation of the conventional explicit FDTD method instead of from Maxwell's equations or conventional ADI-FDTD method. Moreover, the unconditional stability of the leapfrog ADI-FDTD method for lossy media is only numerically verified.

In this paper, the leapfrog ADI-FDTD method for lossy media is first derived from the conventional ADI-FDTD method. Then its unconditional stability is analytically proved though a method that combines the von Neumann method with the Jury criterion. At last, its accuracy and unconditional stability are studied through numerical experiments.

2. LEAPFROG ADI-FDTD FORMULATION FOR LOSSY MEDIA

The time-dependent Maxwell's curl equations in a linear, lossy, and non-dispersive medium with permittivity [epsilon], permeability [mu], and electric conductivity [sigma] can be written as

[epsilon] dE/dt = [(A - B) H - aE] (1a)

[mu] dH/dt = (B - A) E (1b)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, for sub-step #1 of the ADI-FDTD method:

[epsilon][E.sup.n+1/2] = [epsilon][E.sup.n] + [DELTA}t/2 (A[H.sup.n+1/2] - B[H.sup.n] - [sigma][E.sup.n+[m.sub.1]])(2a)

[mu][H.sup.n+1/2] = [mu][H.sup.n] + [DELTA]t/2 (B[E.sup.n+1/2] - A[E.sup.n]). (2b)

For sub-step #2:

[epsilon][E.sup.n+1] = [epsilon][E.sup.n+1/2] + (A[H.sup.n+1/2] - B[H.sup.n+1] - [sigma][E.sup.n+[m.sub.2]]) (3a)

[mu][H.sup.n+1] = [mu][H.sup.n+1/2] + [DELTA]t/2 (B[E.sup.n+1/2] - A[E.sup.n+1]). (3b)

where [m.sub.1] and [m.sub.2] in (2a) and (3a) are the time indices within one time step. Following the same derivative process as given in [4], by substituting (2b) into (2a), we have

[epsilon][E.sup.n+1] = [epsilon][E.sup.n] + [DELTA]t/2 (A[H.sup.n] + [DELTA]t/2 A[[mu].sup.-1] B[E.sup.n+1/2] - [DELTA]t/2 A[[mu].sup.-1] A[E.sup.n] - B[H.sup.n] - [DELTA]t/2 [sogma][E.sup.n+[m.sub.1]]. (4)

Replacing n with n - 1 in (3a) and (3b), and then substituting (3b) back into (3a), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Add (4) and (5) on their both sides, the leapfrog ADI-FDTD equation for electric field E is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

With similar procedure, the equation for the magnetic field H can be obtained as follows. Rewrite (2a) as

[epsilon][E.sup.n] = [epsilon][E.sup.n+1/2] - [DELTA]t/2 (A[H.sup.n+1/2] - B[H.sup.n] - [sigma][E.sup.n+[m.sub.1]])). (7)

Substitute (7) into (2b), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Add (8) and (9) on their both sides, the leapfrog ADI-FDTD equation for magnetic field H is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

It is noted that (6) and (10) are the leapfrog ADI-FDTD equations for lossy media which is one-step and arbitrary order in space. Like in [9], various values for the ADI-FDTD method can be chosen for the values of [m.sub.1] and [m.sub.2], e.g., [m.sub.1] =0, [m.sub.2] = 0.5 for the backward-backward method; [m.sub.1] = 0, [m.sub.2] = 1 for the backward-forward method; [m.sub.1] = 0.5, [m.sub.2] = 0.5 for the forward-backward method; and [m.sub.1] = 0.5, [m.sub.2] = 1 for the forward-forward method. We can see that only for [m.sub.1] = 0.5, [m.sub.2] = 0.5 the cross term in (10) can be eliminated. Then we can obtain the leapfrog ADI-FDTD method for lossy media as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11b)

For clarity, the update equations for Ex and Hx are as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12b)

Other field equations can be obtained with cyclic permutations of {x,y,z}. It should be noted again that in (12a) and (12b), the difference approximations are only used for the time. Therefore, the proposed method is in arbitrary order in space, and the spatial derivatives can be replaced with their finite-difference counterparts of a chosen order in actual computations. For example, if we chose the center second-order difference in space domain in an isotropic homogeneous media characterized by permittivity e, permeability //, and electric conductivity [sigma], (12a) and (12b) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13b)

where i, j and k are the spatial indices in the x-, y- and z-directions, respectively. Other field equations can be obtained with cyclic permutations of {x, y, z}.

3. STABILITY ANALYSIS OF THE LEAPFROG ADI-FDTD METHOD FOR LOSSY MEDIA

To analyze the stability of the leapfrog ADI-FDTD method for lossy media, the von Neumann method combining with the Jury criterion is employed [12]. The von Neumann method mainly consists of considering a Fourier series expansion of the field components at the mesh nodes at a given time as

[f.sub.n](i, j, k) = [f.sub.0][Z.sup.n] exp [J(i[DELTA][xk.sub.x] + j[DELTA][yk.sub.y] + k[DELTA][zk.sub.z])] (14)

where J is the imaginary unit, [f.sub.0] the complex amplitude, Z the amplification factor, and [k.sub.x], [k.sub.y] and [k.sub.z] are the wavenumbers in the x-, y- and z-directions. To determine the value of [absolute value of (Z)], field solutions of the form (14) are substituted into the difference equations of the leapfrog ADI-FDTD method for lossy media. This leads to a characteristic polynomial in Z as

S (Z) = [N.summation over (i=0)] [a.sub.i][Z.sup.i] (15)

The condition for stability can be written as [absolute value of ([Z.sub.i])] [less than or equal to] 1.0 where [Z.sub.i] are the roots of S(Z) = 0. A sufficient condition to ensure this is the Jury criterion which only uses the coefficients of S(Z). To do so, a Jury table should be given by

In Table 1, the entries

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where j = 0, 1, ..., N - 1 and k = 0, 1, ..., N - 2.

Jury criterion is that if the conditions below are satisfied,

s(1) > 0, [(-1).sup.N] S(-1) > 0, [a.sub.N] > [absolute value of ([a.sub.0])], [b.sub.N-1] > [absolute value of ([b.sub.0])], [c.sub.N-2] > [absolute value of ([c.sub.0])], ..., [d.sub.2] > [absolute value of ([d.sub.0])]. (17)

then all the roots of S(Z) = 0 are less or equal to unity in magnitude.

Without generosity, for the leapfrog ADI-FDTD method for lossy media, by substituting (14) into the second-order differential leapfrog ADI-FDTD Equation (13), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18a)

For a nontrivial solution of (18), the determinant of coefficient matrix should be zero. Therefore, with the help of the software Mathematica 8.0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Here, for simplicity, we divide S(Z) into three different polynomials [S.sub.1](Z), [S.sub.2](Z), and [S.sub.3](Z).

It is obvious that the root for [S.sub.1] (Z) = 0 is unity.

For [S.sub.2](Z), by applying Jury criterion, we have

[S.sub.2](1) = 4 ([Q.sub.y] [W.sup.2.sub.x] + [Q.sub.z] [W.sup.2.sub.y] + [Q.sub.x][W.sup.2.sub.z]) > 0 (20a)

[S.sub.2](-1) = 4[epsilon][mu] + 4[W.sup.2.sub.x][W.sup.2.sub.y][W.sup.2.sub.z] / ([[epsilon].sup.2] [[mu].sup.2]) > 0 (20b)

[a.sub.2] = [a.sub.0] = [epsilon] [mu] [Q.sub.x][Q.sub.y][Q.sub.z] > 0. (20c)

Note that (20b) is obtained by substitution (18c) into the right hand of [S.sub.2](-1). Equation (20c) is the critical condition of the Jury criterion. This means that both the roots of [S.sub.2](Z) = 0 is unity in magnitude. For [S.sub.3](Z), Jury criterion is applied again, we have

[S.sub.3](1) = 8[Q.sub.L][W.sup.2.sub.x] > 0 (21a)

[(-1).sup.3][S.sub.3](-1) = 8[epsilon][mu] + 8[W.sup.2.sub.x][W.sup.2.sub.y][W.sup.2.sub.z]/ ([[epsilon].sup.2] [[mu].sup.2]) > 0 (21b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21d)

Note again that (21b) is obtained by substitution (18c) into the right hand of [S.sub.3](-1). So the magnitudes of the three roots of [S.sub.3](Z) = 0 are less than unity.

As a result, all the six roots of S(Z) = 0 are not lager than unity. Therefore, the proposed leapfrog ADI-FDTD method for lossy media is unconditionally stable.

4. NUMERICAL RESULTS AND DISCUSSION

To prove the unconditional stability and accuracy of the proposed leapfrog ADI-FDTD method, a lossy material-filled PEC cavity with Yee grid cells of 50 x 30 x 9 is calculated as in Fig. 1.

A line current source centred at the domain from bottom to top along the z-direction was used to excite the EM field. The current source is given by

[J.sub.z] =exp [-4[pi][(t-[t.sub.0]).sup.2] / [[TAU].sub.2]] (22)

where [DELTA]x = [DELTA]y = [DELTA]z = [DELTA] = 2 mm, [DELTA]t = [DELTA]/ [square root of (3c)] * CFLN, [TAU] = 150[DELTA]t/CFLN, and [t.sub.0] = 2[TAU], CFLN being the CFL number. We set the observation point five cells away from the centre of the computational domain along the x-direction.

Figure 2 shows [E.sub.z] component versus time at the observation point obtained with (i) leapfrog ADI-FDTD method with CFLN = 1 and CFLN = 10, (ii) conventional ADI-FDTD with CFLN = 10, (iii) FDTD with CFLN = 1. We choose the permittivity e and permeability [mu] as the values in vacuum. The electric conductivity a is chosen as 0.01 S/m. It can be seen that the result obtained by CFLN = 1 for the leapfrog ADI-FDTD method agrees well with that obtained by FDTD; the errors are increasing slightly with the increasing of CFLN from 1 to 10; for CFLN = 10, the leapfrog ADI-FDTD and conventional ADI-FDTD result have almost the same errors. To check the stability, many more experiments were carried out with different electric conductivities and CFLNs. We use 100 million iterations for each simulation. The electric conductivities are changed from 0 to [10.sup.6] S/m and CFLN from 1 to [10.sup.3]. It is found that the recorded [E.sub.z] always decreases as the time progresses. Therefore, the stability of the proposed method is numerically verified.

Table 2 shows the comparisons of the CPU time and memory used by the conventional FDTD with CFLN = 1, conventional ADI FDTD with CFLN = 1 and 10, proposed leapfrog ADI-FDTD with

CFLN = 1 and 10. The simulation platform is Lenovo PC with Intel Dual Core i3 M330 of 2.13 GHz and RAM of 4 GB; the source code was written in Fortran language. It can be seen that the proposed method used the same memory as that of FDTD. This is because, unlike the conventional ADI-FDTD method, the leapfrog ADI-FDTD method has only one-step and with no mid-time EM fields needed to be stored. As for the CPU time consumed, the proposed method with CFLN = 10 uses the least.

5. CONCLUSION

A one-step leapfrog ADI-FDTD method for lossy media has been presented in this paper. This method is derived from the conventional ADI-FDTD method. Its unconditional stability has been analytically proven though a method that combines the von Neumann method with the Jury criterion. In addition, its unconditional stability and accuracy are verified through numerical experiments.

ACKNOWLEDGMENT

This work was supported by the Tianjin Research Program of Application Foundation and Advanced Technology, China, under Grant 12JCYBJC10500.

Received 2 November 2012, Accepted 6 May 2013, Scheduled 11 May 2013

REFERENCES

[1.] Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, 3rd Editon, Artech House, Norwood, MA, 2005.

[2.] Namiki, T., "A new FDTD algorithm based on alternating-direction implicit method," IEEE Trans. Microw. Theory Tech., Vol. 47, No. 10, 2003-2007, Oct. 1999.

[3.] Zheng, F., Z. Chen, and J. Zhang, "Toward the development of a three dimensional unconditionally stable finite-different time-domain method," IEEE Trans. Microwave Theory Tech., Vol. 48, No. 9, 1550-1558, Sep. 2000.

[4.] Cooke, S. J., M. Botton, T. M. Antonsen, and B. Levush, "A leapfrog formulation of the 3D ADI-FDTD algorithm," Int. J. Numer. Model, Vol. 22, No. 2, 187-200, 2009.

[5.] Yang, S. C., Z. Chen, Y. Yu, and W. Y. Yin, "An unconditionally stable one-step arbitrary-order leapfrog ADI-FDTD method and its numerical properties," IEEE Trans. Antennas Propag., Vol. 60, No. 4, 1995-2003, Apr. 2012.

[6.] Gan, T. H. and E. L. Tan, "Stability and dispersion analysis for three-dimensional (3-D) leapfrog ADI-FDTD method," Progress In Electromagnetics Research M, Vol. 23, 1-12, Jan. 2012.

[7.] Heh, D. Y. and E. L. Tan, "Dispersion analysis of FDTD schemes for doubly lossy media," Progress In Electromagnetics Research B, Vol. 14, 177-192, 2010.

[8.] Velarde, L. F., J. A. Pereda, A. Vegas, and O. Gonzalez, "A weighted-average scheme for accurate FDTD modeling of electromagnetic wave propagation in conductive media," IEEE Antennas Wireless Propag. Lett., Vol. 3, 302-305, 2004.

[9.] Pereda, J. A., A. Grande, O. Gonzalez, and A. Vegas, "The 1D ADI-FDTD method in lossy media," IEEE Antennas Wireless Propag. Lett., Vol. 7, 477-480, 2008.

[10.] Heh, D. Y. and E. L. Tan, "Unified efficient fundamental ADI-FDTD schemes for lossy media," Progress In Electromagnetics Research B, Vol. 32, 217-242, 2011.

[11.] Gan, T. H. and E. L. Tan, "Unconditionally stable leapfrog ADI-FDTD method for lossy media," Progress In Electromagnetics Research M, Vol. 26, 173-186, 2012.

[12.] Xiao, F. and X. H. Tang, "Stability and numerical dispersion analysis of a fourth-order accurate FDTD method," IEEE Trans. Antennas Propag., Vol. 54, No. 9, 2525-2530, Sep. 2006.

Jian-Yun Gao (1), * and Hong-Xing Zheng (2)

(1) Department of Basic Courses, Tianjin Vocational Institute, Tianjin 300410, China

(2) Institute of Antenna and Microwave Techniques, Tianjin University of Technology and Education, Tianjin 300222, China

* Corresponding author: Jian-Yun Gao (jianyun_gao@163.com).

Table 1. Jury table.

1       [a.sub.N]    [a.sub.N-1]   [a.sub.N-2]   ...
2       [a.sub.0]     [a.sub.1]     [a.sub.2]    ...
3      [b.sub.N-1]   [b.sub.N-2]   [b.sub.N-3]   ...
4       [b.sub.0]     [b.sub.1]     [b.sub.2]    ...
5      [c.sub.N-2]   [c.sub.N-3]   [c.sub.N-4]   ...
6       [c.sub.0]     [c.sub.1]     [c.sub.2]    ...
[??]      [??]          [??]          [??]       [??]
2N-3    [d.sub.2]     [d.sub.1]     [d.sub.0]     0

1       [a.sub.2]     [a.sub.1]    [a.sub.0]
2      [a.sub.N-2]   [a.sub.N-1]   [a.sub.N]
3       [b.sub.1]     [b.sub.0]        0
4      [b.sub.N-2]   [b.sub.N-1]       0
5       [c.sub.0]         0            0
6      [c.sub.N-2]        0            0
[??]      [??]          [??]         [??]
2N-3        0             0            0

Table 2. Comparison of the CPU time and memory used by the
conventional FDTD method, conventional ADI-FDTD method, and
proposed leapfrog ADI-FDTD method.

Method      CFLN   Number of   CPU        Memory
                   steps       Time (s)   (MB)

leapfrog    1      2000        25         3.7
ADI-FDTD    10     200         2.6        3.7
ADI-FDTD    10     200         3.3        4.1
FDTD        1      2000        9.4        3.7
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Title Annotation:alternating-direction-implicit finite-difference time-domain
Author:Gao, Jian-Yun; Zheng, Hong-Xing
Publication:Progress In Electromagnetics Research Letters
Article Type:Abstract
Geographic Code:9CHIN
Date:May 1, 2013
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