Research Article One-Step Leapfrog LOD-BOR-FDTD Algorithm with CPML Implementation

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1 Antennas and Propagation Volume 2016, Article I , 8 pages Research Article One-Step Leapfrog LO-BOR-FT Algorithm with CPML Implementation Yi-Gang Wang, 1 Yun Yi, 1 Bin Chen, 1 Hai-Lin Chen, 1 Kang Luo, 1 and Run Xiong 2 1 PLA University of Science and Technology, Nanjing, Jiangsu , China 2 Engineering Academy of PLA, Xuhou, Jiangsu , China Correspondence should be addressed to Yun Yi; yiyun rachel@sohu.com Received 5 January 2016; Revised 12 March 2016; Accepted 14 April 2016 Academic Editor: Marta Cavagnaro Copyright 2016 Yi-Gang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An unconditionally stable one-step leapfrog locally one-dimensional finite-difference time-domain (LO-FT) algorithm towards body of revolution (BOR) is presented. The equations of the proposed algorithm are obtained by the algebraic manipulation of those used in the conventional LO-BOR-FT algorithm. The equations for -direction electric and magnetic fields in the proposed algorithm should be treated specially. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LO-BOR-FT algorithm. Moreover, the convolutional perfectly matched layer (CPML) is introduced into the one-step leapfrog LO-BOR-FT algorithm. The equation of the one-step leapfrog CPML is concise. Numerical results show that its reflection error is small. It can be concluded that the similar CPML scheme can also be easily applied to the one-step leapfrog LO-FT algorithm in the Cartesian coordinate system. 1. Introduction The body of revolution finite-difference time-domain (BOR- FT) algorithm is very efficient in analying electromagnetic problems towards rotationally symmetric structures [1, 2]. It has been widely used in modeling electromagnetic pulse effects, electromagnetic wave scattering, subsurface interface radar, optical lenses, guided waves, and so on [1]. However, the time step sie of the conventional BOR-FT algorithm is strictly limited by the Courant-Friedrichs-Lewy (CFL) condition [1]. To remove the stability limit on the time step sie of the BOR-FT algorithm and improve the efficiency, some unconditionally stable schemes such as the alternatingdirection implicit (AI) BOR-FT [3], the locally onedimensional (LO) BOR-FT [4], and the weighted Laguerre polynomials (WLP) BOR-FT [5] algorithms have been proposed. The WLP-BOR-FT algorithm needs tosolvealargesparsematrix,soitisnotsoapplicablefor large computational domain [5]. The LO-BOR-FT algorithm and the AI-BOR-FT algorithm show comparable accuracy, and the LO-BOR-FT algorithm shows a little higher computational efficiency [4]. In the conventional AI- BOR-FT and LO-BOR-FT algorithms, the equations for one full time step are split into two subtime steps; as a result, their computational expenditures are increased [3, 4]. Recently, the one-step leapfrog AI-FT algorithm which eliminates the midtime step successfully has been proposed anddeveloped[6 11].ItmakesthesimulationwiththeAI- FT algorithm more efficient. The application of the onestep leapfrog AI-FT algorithm to BOR has also been proposed [12]. In fact, the parallel improvement can also be made for the conventional LO-BOR-FT algorithm. Recently, an unconditionally stable one-step leapfrog LO-FT algorithm was proposed [13]. In the algorithm, theequationsareobtainedbythemanipulationofthoseused in the conventional LO-FT algorithm. The resultant electric and magnetic field equations are interlaced half a time step apart and no subtime steps are involved. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LO-FT algorithm [13].

2 2 Antennas and Propagation In this work, the one-step leapfrog LO-FT algorithm for BOR is developed, called one-step leapfrog LO-BOR- FT algorithm. In the proposed algorithm, the -direction electric and magnetic field components are dealt with differently. Moreover, the convolutional perfectly matched layer (CPML) [14] is introduced to the one-step leapfrog LO-BOR-FT algorithm. The equations of the one-step leapfrog CPML are concise. The remainder of the paper is organied as follows. In Section 2, the equations of the one-step leapfrog LO-BOR- FT algorithm are presented and some discussions about the algorithm are made. In Section 3, the CPML is developed for the proposed algorithm. To assess the proposed algorithm and its CPML, numerical examples are given in Section 4. Finally, conclusions are made in Section Formulations and iscussions 2.1. Equations for Off-Axis Cells. The equations for the conventional LO-BOR-FT algorithm can be expressed as [4] A 12H n+1/2 = E n A 12H n H n+1/2 2μ A 21 = H n 2μ A 21E n for the first subtime step and E n+1 B 12H n+1 = B 12H n+1/2 H n+1 2μ B 21E n+1 = H n+1/2 2μ B 21 for the second subtime step. Here, ± m A 12 =, [ () ] [ ] A 21 =, [ m ] [ ] B 12 =, [ ± m ] [ ] (1a) (1b) (2a) (2b) m B 21 = [ () ] [ ] are the matrices that contain the spatial differential operators, E =[E E E ] T and H =[H H H ] T are the electric and magnetic field vectors, m isthemodenumber, is the permittivity, andμ is the permeability. Following the similar procedure used in [13, 15, 16], one can obtain the following electric field equations for the onestep leapfrog LO-BOR-FT algorithm with only algebraic manipulations of (1a), (1b), (2a), and (2b): Ẽ n1/2 = E n1/2 B 12h n ( I 2 2 8μ A 12A 21 ) = Ẽ n1/2 A 12H n = Ẽ n1/2, (3) (4a) (4b) (4c) where I is a 3 3unit matrix, Ẽ n1/2 and are auxiliary variables, and = + E n. Similarly, one can obtain the equations for the magnetic fields H n = H n 2μ A 21 (5a) ( I 2 2 8μ B 21B 12 ) h n+1 = H n 2μ B 21 H n+1 = h n+1 H n, (5b) (5c) where H n and h n+1 are auxiliary variables and h n+1 = H n+1 + H n+1/ Equations for On-Axis E Field. The on-axis E field component cannot be obtained by using (4a), (4b), and (4c) directly.notethattheon-axise field component is ero for m =0, so one should only deal with it for m=0.theequation for the on-axis E field component in the conventional LO- BOR-FT algorithm is [4] (0, k + 1 2Δt ) 2 Δ Hn+1/2 ( 1 2,k+ 1 2 ) =E n (0, k + 1 2Δt )+ 2 Δ Hn (1 2,k+ 1 2 ) for the first subtime step and (6a) E n+1 (0, k )=En+1/2 (0, k ) (6b)

3 Antennas and Propagation 3 forthesecondsubtimestep.moreover,onecanobtainthe equation for H n+1/2 (1/2, k + 1/2) in the conventional LO- BOR-FT algorithm from (1b) H n+1/2 ( 1 2,k+ 1 2 ) 2μ (1, k + 1/2) (0, k + 1/2) Δ =H n (1 2,k+ 1 2 ) 2μ E n (1,k+1/2) En (0, k + 1/2). Δ Applying the similar procedure used in Section 2.1, one can obtain the following equations for the on-axis E field component by using (6a), (6b), and (7): ( μδ 2 )en+1/2 (0, k ) Δt2 μδ 2 en+1/2 + 2Δt Δ Hn (1 2,k+ 1 2 ) (1, k )=En1/2 (0, k ) (7) (8a) (0, k )=en+1/2 (0, k ) E n1/2 (0, k + 1 (8b) 2 ). Unlike the equations for the off-axis E field components, no auxiliary variable E is involved here Equations for H n+1 (1/2, k). The equations for H n+1 (1/2, k) should also be treated specially. With the algebraic manipulation of the relative difference equations used in the conventional LO-BOR-FT algorithm [4], one can obtain the following equations for H n+1 (1/2, k): H n (1 2,k) =Hn (1 Δt,k) + 2 μ ( m Δ ) en+1/2 ( 1,k) (9a) 2 ( μΔ 2 )hn+1 ( 1 Δt2,k) 2 = H n (1 Δt,k) 2 μδ En+1/2 4μΔ 2 hn+1 (1, k) ( 3 2,k) (9b) H n+1 ( 1 2,k)=hn+1 ( 1 2,k) H n (1,k). (9c) Some iscussions about the Proposed Algorithm. It can be seen that all the equations used in the proposed algorithm are obtained from those used in the conventional LO-BOR- FT algorithm and only algebraic manipulations are made. Therefore, one can conclude that the proposed algorithm preserves the properties of the conventional LO-BOR- FT algorithm. In terms of memory, the variables used in the conventional LO-BOR-FT algorithm are, H n+1/2,and E n in the first subtime step, which can also be reused in the second subtime step. In the proposed algorithm, Ẽ n1/2, E n1/2,and can occupy the same memory space. There are similar situations to H n, H n,andh n+1 and and h n+1. As a result, the proposed algorithm consumes the same amount of memory as the conventional LO-BOR-FT algorithm. There are four tridiagonal equations in one full time step, for both the conventional LO-BOR-FT algorithm and the proposed algorithm. So the floating point operations at the left-hand sides of the equations are the same for the two algorithms. However, the numbers of the multiplications/divisions (M/) and the additions/subtractions (A/S) at the right-hand sides of the equations are different. Table 1 shows the count of the M/ and A/S at the right-hand sides of the equations in two algorithms for modes m = 0 and m =0, respectively. Obviously, the proposed algorithm needs lessfloatingpointoperations,soonecanconcludethatithasa higher computational efficiency. Note that all the coefficients oftheequationsareprecomputedandstoredhere,andspecial treatments for the components that lie on or near the axis are not considered. In the proposed algorithm, tridiagonal implicit equations for the magnetic field components are involved, so one should pay special attention to handling the perfectly electrically conducting (PEC) boundary, since there will be out-ofdomain magnetic field components. To address this issue, one can resort to the image theory [17]. Such treatments are often used in handling the PEC resonator, PEC scatterer, PEC wall of the perfectly matched layer (PML), and so on. 3. CPML Implementation for the Proposed Algorithm To solve open region problems efficiently, the CPML for the proposed one-step leapfrog LO-BOR-FT algorithm is developed. The coordinate-stretching variables for the CPML are defined as σ s =κ + α + jω, One can also define s =κ + s = =κ + σ α + jω. (10) σ α + jω, (11) where s =(1/)[ 0 + (κ 0 +(σ /(α + jω)))d ] and 0 represents the interface between FT and PML domains [18]. For simplicity, the CPML equations for the E field component are given here. In fact, the CPML equations for

4 4 Antennas and Propagation Table 1: The floating point operations at the right-hand sides of the equations for two algorithms. Mode Conventional LO-BOR-FT One-step leapfrog LO-BOR-FT m=0 m=0 m=0 m=0 M/ A/S M/ + A/S the other field components can be obtained in a similar manner. According to the definition in (10) and (11), one can write the CPML equations of the conventional LO-BOR- FT algorithm in the following form: n+1/2 ( 1 H κ + 1 κ H n+1/2 e e )=E n ( 1 n H κ + 1 κ H n +ψ n e +ψn e ) (12a) Then(12a)and(12b)canberewrittenas h n+1/2 ( 1 h κ + n+1/2 e )=2E n n+1/2 + 1 κ h n+1/2 + n+1/2 e (15a) 2μ ( 1 κ + n+1/2 h )=2Hn. (15b) H n+1/2 2μ ( 1 κ 2μ ( 1 κ E n +ψn h ) for the first subtime step and E n+1 H n+1 + = ( 1 H n+1 κ h )=Hn +ψ n+1 e ) ( 1 H n+1/2 κ 2μ ( 1 κ E n+1 +ψ n+1 h ) =H n+1/2 e ) 2μ ( 1 κ h ) (12b) (13a) (13b) In the second subtime step, one can define e n+1 h n+1 =E n+1 =H n+1 Then(13a)canberewrittenas e n H n+1/2 (16a) (16b) n+1 e =ψn+1 e +ψn+1/2 e. (16c) ( 1 h n+1 κ The former time steps of (16a) and (17) are +n+1 e )=2En+1/2. (17) for the second subtime step. In the first subtime step, one can define h n+1/2 n+1/2 e n+1/2 e n+1/2 h = +E n (14a) =H n+1/2 +H n (14b) =ψn+1/2 e +ψn e (14c) =ψ n+1/2 e +ψ n e (14d) =ψ n+1/2 h +ψ n h. (14e) e n + Substituting (19) into (18), one can obtain E n1/2 e n =En +En1/2 (18) ( 1 κ h n +n e )=2En1/2. (19) ( 1 κ h n +n e ) =En. (20)

5 Antennas and Propagation 5 Substituting (15b) into (15a), one can obtain 1 1 κ Hn 1 1 Δt κ 2μ ( 1 κ 1 κ H n 1 Δt κ 2μ ( 1 κ (n+1/2 e +n+1/2 e )=2E n. + n+1/2 h ) + n+1/2 h ) (21) ( 1 κ H n +ψn e )+Δt 1 1 Δt κ 1 Δt κ 2μ n1/2 h h. (24) 2μ n1/2 The following equation can be obtained from (14a) and (20): = E n1/2 + ( 1 κ h n +n e ). (25) To simplify the equations further, one can introduce an auxiliary variable E n1/2. As a result, the final equations for E canbereformulatedas Substituting (20) into (21), the following equation can be obtained: =2E n1/2 1 1 Δt κ 2μ ( 1 κ 1 Δt κ 2μ ( 1 κ + n+1/2 h ) + n+1/2 h ) ( 1 κ h n +n e ) 1 1 κ Hn 1 κ H n (n+1/2 e +n+1/2 e ). Some of the auxiliary variables in (22) are approximated as n+1/2 h n+1/2 e n+1/2 e (22) = n1/2 h +O(Δt) (23a) =ψn+1/2 e +ψn e =2ψn e +O(Δt) (23b) =ψ n+1/2 e +ψ n e =2ψn e +O(Δt). (23c) Ignoring the high order terms of Δt in (23a), (23b), and (23c), then(22)canbeapproximatedas =2E n1/2 1 1 Δt 1 κ 2μ κ 1 Δt 1 κ 2μ κ ( 1 κ 1 Hn +ψn e ) ( 1 κ h n +n e ) E n1/2 =E n1/2 1 2 en+1/ μ κ κ μ κ κ ( 1 κ 1 Hn +ψn e ) ( 1 κ h n +n e ) = E n1/2 ( 1 κ H n +ψn Δt2 1 1 e )+ 8μ κ 2 1 8μ κ n1/2 h = n1/2 h (26a) (26b) E n1/2. (26c) Some of the auxiliary variables in (26a), (26b), and (26c) can be solved by n e =b n1 e +a ( h n ) ψ n e =b ψ n1 e +a ( H ) n (27a) (27b) ψ n e =b ψ n1 e +a ( H n ), (27c) where b ς =e (σ ς/κ ς +α )Δt/ ς and a ς =(σ ς /(κ ς (σ ς +κ ς α ς )))(b ς 1), ς=,,. In addition, the auxiliary variable n1/2 h in (26b) has been solved in updating H n, so it can be used directly here. It is obvious that the number of the convolutional auxiliary variables for the proposed CPML is the same as that of the CPML for the conventional BOR-FT algorithm.

6 6 Antennas and Propagation BOR-FT Table 2: Comparison of the computational results with different algorithms. Conventional LO-BOR-FT One-step leapfrog AI-BOR-FT One-step leapfrog LO-BOR-FT CFLN Iterations Results, GH Relative error 0.08% 0.08% 0.14% 0.87% 0.08% 0.14% 0.87% 0.08% 0.14% 0.87% Time, s Consequently, one can say that the equations of proposed CPML are concise. When κ=1and σ=0, the equations of the proposed CPML can be simplified to be the normal equations of the proposed LO-BOR-FT algorithm. Similarly, since the coordinate stretching along the -direction only exists in the radial outer boundary, that is, κ =κ =1and σ =σ =0on (or near) the axis, the CPML equations for (0, k + 1/2) and H n+1 (1/2, k) are still (8a), (8b), (9a), (9b), and (9c). In the Cartesian coordinate system, the coordinate stretchingalongeachaxisissimilartothatalongthedirection in the cylindrical coordinate system. So it can be concluded that the CPML for the LO-FT algorithm in the Cartesian coordinate system [13] can be implemented by applying the similar procedure. 4. Numerical Results and iscussions A resonant cavity and a scattering example are provided to validate the one-step leapfrog LO-BOR-FT algorithm and its CPML Cavity Resonator. The TE 011 mode of a PEC cylindrical cavity resonator is computed, which is frequently used as a benchmark [3, 4]. The radius and height of the cavity are cm and cm, respectively. Its theoretical resonant frequency is GH [3, 4]. The computational domain is divided into 50() 100() cells, with Δ = cm and Δ = cm. The cavity is computed with the conventional BOR-FT algorithm, the conventional LO- BOR-FT algorithm, the one-step leapfrog AI-BOR- FT algorithm, and the proposed algorithm, respectively. Atimehistoryof52.73nsissimulated. Table 2 shows the computational results with different algorithms. Note that CFLN = Δt/Δt CFL and Δt CFL is the maximum time step sie that satisfies the CFL condition for the conventional BOR-FT algorithm [1], which is Δt CFL = shere.itcanbeseenfromtable2thatthe one-step leapfrog LO-BOR-FT algorithm obtains the same numerical results as the conventional LO-BOR-FT algorithm. Moreover, the numerical errors of two algorithms increase as the time step sies increase. In terms of the computational time, the proposed algorithm consumes about 26.5%, 25.8%, and 25.0% less time than the conventional LO-BOR-FT algorithm when CFLN is chosen to be 1, 4, and 8, respectively. The two unconditionally stable algorithms use the same amount of memory, which is 1.06 MB. Therefore, one can say that the proposed algorithm preserves the properties of the conventional LO-BOR-FT algorithm and obtains a higher computational efficiency. The proposed algorithm also shows the same accuracy as that of the onestep leapfrog AI-BOR-FT algorithm but obtains a little higher computational efficiency, as shown in Table 2. Note that the comparison among the conventional LO-BOR- FT algorithm, the one-step leapfrog AI-BOR-FT algorithm, and the proposed algorithm was performed only in the case of the resonant cavity and that further analysis could be performed on other situations to rule out the possibility that the results could be linked to the particular case considered. Since the TE 011 mode is computed, the terms including multiplicative factor m (mode number) in theequationsareallero.inthesimulation,thesetermsare notcomputed.allofthecalculationsareperformedonan Intel Core 2QuadCPUQ6600@2.40GHmachineand the Fortran codes are used Scattering Example. TovalidatetheproposedCPML for the one-step leapfrog LO-BOR-FT algorithm, the scatteredfieldfromapeccylinderiscomputed.theradius and height of the cylinder are 20 cm and 40 cm, respectively. The oblique incident plane wave with incident angle θ i =45 is introduced by the TF/SF boundary conditions, which are placed at = 25cm and = ±25cm. The incident plane wave is parallel polaried, as shown in Figure 1. The gray domain in Figure 1 denotes the PEC cylinder. The Gaussian pulse E 0 cos(2πf 0 t)e 4π((tt 0)/τ) 2 with E 0 = 1000 V/m, f 0 = 0.3 GH, τ = 2.2 ns, and t 0 = 1.25/f 0 is used as the incident electric field E i. The small computational domain is composed of 40() 80() cells, containing the 10 PML layers in the outer boundary, which are denoted by the oblique-linedomain as shown in Figure 1. According to the characteristic of the problem, the mode number ranges from 0 to 11. The uniform grids are chosen to be Δ = Δ = 1 cm. Therefore, the maximum time step sie for the conventional BOR-FT [1] is Δt CFL = s in the example. The observation point is placed at grid point Q(29, 29), whichisnearthe corner of the scattered field domain. The scattered field is computed with the conventional BOR-FT algorithm and the proposed algorithm, respectively, both with the CPML absorbing boundary conditions.

7 Antennas and Propagation 7 E i O θ i k H i TF/SF boundaries E (V/m) Q Figure 1: Illustration of the scattering example with oblique incidence BOR-FT CFLN = Time (ns) CFLN =4 CFLN =8 Figure 2: E fields at the observation point Q with the conventional BOR-FT and the proposed algorithm. The parameters of the CPML are chosen according to the following expressions [18]: σ ς = σ ς max ςς 0 p δ p σ ς opt = (p + 1) 150πΔς (28a) (28b) κ ς =1+κ ςς 0 p ς max δ p, (28c) where ς=,,δisthe thickness of the PML, ς 0 is the interface between the FT and PML grids, and p=4is the order of the polynomial. In the computing, the optimied parameters arechosentobeκ ς max =3, σ max = 0.4σ opt,andα = To validate the one-step leapfrog LO-BOR-FT algorithm further, the scattered field is computed via large computational domain first. In this case, the scattered field at the observation point Q is without reflection, so one can make a clear comparison. Figure 2 shows the E fields obtained by the two algorithms via large computational domains. For the proposed algorithm, different CFLNs are used, where CFLN = Δt/Δt CFL. It can be seen that the results of the proposed algorithm agree well with those of the conventional BOR-FT algorithm. Therefore, the proposed algorithm is validated further. Meanwhile, one can see that the numerical error of the proposed algorithm increases as the time step sie grows. Figure 3 shows the reflection errors of the CPMLs for the conventional BOR-FT algorithm and the proposed algorithm, which are obtained by the E fields at the observation point Q. The reflection error is defined as R db =20log 10 ( E pml (t) E ref (t) max E ), (29) ref (t) Relative reflection error (db) CPML CFLN = Time (ns) CFLN =4 CFLN =8 Figure 3: Relative reflection errors of the CPML for the conventional BOR-FT and the proposed CPML for the one-step leapfrog LO-BOR-FT. where E pml (t) is the result obtained by the small computationaldomainasshowninfigure1.e ref (t) is the numerical result via large computational domain. For the proposed algorithm, E ref (t) is computed separately for each CFLN. It canbeseenfromfigure3thatthecpmlfortheone-step leapfrog LO-BOR-FT algorithm shows good performance. Moreover, the reflection error of the one-step leapfrog CPML increases as the time step sie grows. When CFLN =8, the maximum reflection error is about 40 db.

8 8 Antennas and Propagation 5. Conclusion In this paper, the one-step leapfrog LO-BOR-FT algorithm is presented. It obtains a higher computational efficiency while keeping the properties of the conventional LO-BOR-FT algorithm. In addition, the CPML for the proposed algorithm is implemented. The numerical results show the performances of the one-step leapfrog LO-BOR- FT algorithm and its CPML. It can be concluded that, by applying the similar procedure, the CPML for the LO- FTalgorithmintheCartesiancoordinatesystemcanbe implemented. Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1]A.TafloveandS.C.Hagness,Computational Electrodynamics: The Finite-difference Time-domain Method, ArtechHouse, Boston, Mass, USA, 2nd edition, [2] Y. Chen, R. Mittra, and P. Harms, Finite-difference timedomain algorithm for solving maxwell s equations in rotationally symmetric geometries, IEEE Transactions on Microwave Theory and Techniques,vol.44,no.6,pp ,1996. [3] H.-L. Chen, B. Chen, Y. Yi, and.-g. Fang, Unconditionally stable AI-BOR-FT algorithm for the analysis of rotationally symmetric geometries, IEEE Microwave and Wireless Components Letters, vol. 17, no. 4, pp , [4] J.Shibayama,B.Murakami,J.Yamauchi,andH.Nakano, LO- BOR-FT algorithm for efficient analysis of circularly symmetric structures, IEEE Microwave and Wireless Components Letters,vol.19,no.2,pp.56 58,2009. [5] H.-L. Chen, B. Chen, Y.-T. uan, Y. Yi, and.-g. Fang, Unconditionally stable laguerre-based BOR-FT scheme for scattering from bodies of revolution, Microwave and Optical Technology Letters, vol. 49, no. 8, pp , [6]S.J.Cooke,M.Botton,T.M.AntonsenJr.,andB.Levush, A leapfrog formulation of the 3 AI-FT algorithm, in Proceedings of the Workshop on Computational Electromagnetics in Time-omain (CEM-T 07), pp , Perugia, Italy, October [7] S.-C. Yang, Z. Chen, Y. Yu, and W.-Y. Yin, An unconditionally stable one-step arbitrary-order leapfrog AI-FT method and its numerical properties, IEEE Transactions on Antennas and Propagation,vol.60,no.4,pp ,2012. [8] S.-C. Yang, Z. Chen, Y.-Q. Yu, and W.-Y. Yin, The unconditionally stable one-step leapfrog AI-FT method and its comparisons with other FT methods, IEEE Microwave and Wireless Components Letters,vol.21,no.12,pp ,2011. [9] X.-H. Wang, W.-Y. Yin, Y.-Q. Yu, Z.. Chen, J. Wang, and Y. Guo, A convolutional perfect matched layer (CPML) for onestep leapfrog AI-FT method and its applications to EMC problems, IEEE Transactions on Electromagnetic Compatibility, vol.54,no.5,pp ,2012. [10] T. H. Gan and E. L. Tan, Analysis of the divergence properties for the three-dimensional leapfrog AI-FT method, IEEE Transactions on Antennas and Propagation, vol.60,no.12,pp , [11] J.-Y. Gao and H.-X. Zheng, One-step leapfrog AI-FT method for lossy media and its stability analysis, Progress in Electromagnetics Research Letters,vol.40,pp.49 60,2013. [12]Y.-G.Wang,B.Chen,H.-L.Chen,andR.Xiong, Anunconditionally stable one-step leapfrog AI-BOR-FT method, IEEE Antennas and Wireless Propagation Letters, vol.12,pp , [13] J.Wang,B.Zhou,C.Gao,L.Shi,andB.Chen, Leapfrogformulation of the 3- LO-FT algorithm, IEEE Microwave and Wireless Components Letters,vol.24,no.3,pp ,2014. [14] J. A. Roden and S.. Gedney, Convolution PML (CPML): an efficient FT implementation of the CFS-PML for arbitrary media, Microwave and Optical Technology Letters,vol.27,no.5, pp , [15] E. L. Tan, Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods, IEEE Transactions on Antennas and Propagation, vol.56,no.1,pp , [16] J. Shibayama, T. Oikawa, J. Yamauchi, and H. N. Life, Efficient LO-BOR-FT implementation based on a fundamental scheme, IEEE Photonics Technology Letters, vol. 24, no. 11, pp , [17] W. C. Tay and E. L. Tan, Implementations of PMC and PEC boundary conditions for efficient fundamental AI- and LO- FT, Electromagnetic Waves and Applications,vol. 24,no.4,pp ,2010. [18] J. Liu, G. Wan, J. Zhang, and X. Xi, An effective CFS-PML implementation for cylindrical coordinate FT method, IEEE Microwave and Wireless Components Letters, vol.22,no. 6, pp , 2012.

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