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Stability and Dispersion Analysis for Three-Dimensional (3-D) Leapfrog Adi-FDTD Method
Theng Huat Gan;Eng Leong Tan
PIER M , 2012, DOI: 10.2528/PIERM11111803
Abstract: Stability and dispersion analysis for the three-dimensional (3-D) leapfrog alternate direction implicit finite difference time domain (ADI-FDTD) method is presented in this paper. The leapfrog ADI-FDTD method is reformulated in the form similar to conventional explicit FDTD method by introducing two auxiliary variables. The auxiliary variables serve as perturbations of the main fields variables. The stability of the leapfrog ADI-FDTD method is analyzed using the Fourier method and the eigenvalues of the Fourier amplification matrix are obtained analytically to prove the unconditional stability of the leapfrog ADI-FDTD method. The dispersion relation of the leapfrog ADI-FDTD method is also presented.
Unified Efficient Fundamental Adi-FDTD Schemes for Lossy Media
Ding Yu Heh;Eng Leong Tan
PIER B , 2011, DOI: 10.2528/PIERB11051801
Abstract: This paper presents the unified efficient fundamental alternating-direction-implicit finite-difference time-domain (ADI-FDTD) schemes for lossy media. The schemes presented include averaging, forward-forward, forward-backward and novel exponential time differencing schemes. Unifications of these schemes in both conventional and efficient fundamental forms of source-incorporated ADI-FDTD are provided. In the latter, they are formulated in the simplest, most concise, most efficient, and most fundamental form of ADI-FDTD. The unified update equations and implementation of the efficient fundamental ADI-FDTD schemes are provided. Such efficient fundamental schemes have substantially less right-hand-side update coefficients and field variables compared to the conventional ADI-FDTD schemes. Thus, they feature higher efficiency with reduced memory indexing and arithmetic operations. Other aspects such as field and parameter memory arrays, perfect electric conductor and perfect magnetic conductor implementations are also discussed. Numerical results in the realm of CPU time saving, asymmetry and numerical errors as well as specific absorption rate (SAR) of human skin are presented.
High-Order Unconditionally-Stable Four-Step Adi-FDTD Methods and Numerical Analysis
Yong-Dan Kong;Qing-Xin Chu;Rong-Lin Li
PIER , 2013, DOI: 10.2528/PIER12102205
Abstract: High-order unconditionally-stable three-dimensional (3-D) four-step alternating direction implicit finite-difference time-domain (ADI-FDTD) methods are presented. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four sub-procedures. Accordingly, the time step is divided into four sub-steps. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method are used to approximate the spatial differential operators first, and then the uniform formulation of the proposed high-order schemes is generalized. Subsequently, the analysis shows that all the proposed high-order methods are unconditionally stable. The generalized form of the dispersion relations of the proposed high-order methods is carried out. Finally, in order to demonstrate the validity of the proposed methods, numerical experiments are presented. Furthermore, the effects of the order of schemes, the propagation angle, the time step, and the mesh size on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed schemes are lower than that of the traditional ADI-FDTD method.
PML Implementation for ADI-FDTD in Dispersive Media

Wang Yu,Yuan Nai-chang,
王 禹

电子与信息学报 , 2005,
Abstract: Alternating Direction Implicit-Finite Difference Time Domain(ADI-FDTD) is unconditionally stable and the maximum time step size is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step size of ADI-FDTD can be enlarged arbitrarily. In this paper 2D PML implementation is proposed for ADI-FDTD in dispersive media using recursive convolution method. ADI-FDTD formulations for dispersive media can be derived from the simplified Perfectly Matched Layer (PML). Numerical results of ADI-FDTD with PML for dispersive media are compared with FDTD. Good agreement is observed.
Modeling the Interaction of Terahertz Pulse with Healthy Skin and Basal Cell Carcinoma Using the Unconditionally Stable Fundamental Adi-FDTD Method
Ding Yu Heh;Eng Leong Tan
PIER B , 2012, DOI: 10.2528/PIERB11090905
Abstract: This paper presents the application of unconditionally stable fundamental finite-difference time-domain (FADI-FDTD) method in modeling the interaction of terahertz pulse with healthy skin and basal cell carcinoma (BCC). The healthy skin and BCC are modeled as Debye dispersive media and the model is incorporated into the FADI-FDTD method. Numerical experiments on delineating the BCC margin from healthy skin are demonstrated using the FADI-FDTD method based on reflected terahertz pulse. Hence, the FADI-FDTD method provides further insight on the different response shown by healthy skin and BCC under terahertz pulse radiation. Such understanding of the interaction of terahert pulse radiation with biological tissue such as human skin is an important step towards the advancement of future terahertz technology on biomedical applications.
Energy Identities of ADI-FDTD Method with Periodic Structure  [PDF]
Rengang Shi, Haitian Yang
Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.62025
Abstract: In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L2 and H1 norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.
One-step alternating direction implicit FDTD algorithm
Liu Shao-Bin,Liu San-Qiu,

中国物理 B , 2004,
Abstract: In this paper, a novel unconditionally stable alternating direction implicit finite-different time-domain method (ADI-FDTD) called the one-step ADI-FDTD method is presented, where the calculation for one discrete time step is performed using only one procedure, but not the original two sub-updating procedures. Consequently, the proposed one-step ADI-FDTD methods have consumed less computer memory and computation resources and have been faster than the conventional ADI-FDTD methods. We analytically and numerically verified that the new algorithm is unconditionally stable and free from the Courant condition.
An Algorithm of ADI-FDTD and R-FDTD Based on Non-zero Divergence Relationship

Zhou Yong-gang,Xu Jin-ping,Gu Chang-qing,

电子与信息学报 , 2006,
Abstract: In this paper, it is proven that the divergence relationship of electric-field and magnetic-field is non-zero even in charge-free regions, when the electric-field and magnetic-field are calculated with Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) method, and the concrete expression of the divergence relationship is derived. Based on the non-zero divergence relationship, the ADI-FDTD which is unconditionally stable is combined with the Reduced Finite-Difference Time-Domain(R-FDTD). In the proposed method (ADI/R-FDTD), the merit of ADI-FDTD, e.g. increasing time step size and decreasing calculation time, is kept, at the same time, the memory requirement is reduced by 1/3(3-D) or 2/5(2-D) of the memory requirement of ADI-FDTD. Compare to the ADI/R-FDTD based on regular zero divergence relationship, the proposed algorithm is more stable when lager time step size is used. Wave propagation in 2-D free space and the scattered field of a 1-D Frequency Selective Surface(FSS) is simulated by the proposed hybrid method. Compared with ADI-FDTD, perfect agreement of numerical results indicates that ADI/R-FDTD method is correct and efficient.
Dispersion Analysis of FDTD Schemes for Doubly Lossy Media
Ding Yu Heh;Eng Leong Tan
PIER B , 2009, DOI: 10.2528/PIERB09082802
Abstract: This paper presents the 3-D dispersion analysis of finite-difference time-domain (FDTD) schemes for doubly lossy media, where both electric and magnetic conductivities are nonzero. Among the FDTD schemes presented are time-average (TA), time-forward (TF), time-backward (TB) and exponential time differencing (ETD). It is first shown that, unlike in electrically lossy media, the attenuation constant in doubly lossy media can be larger than its phase constant. This further calls for careful choice of cell size such that both wavelength and skin depth of the doubly lossy media are properly resolved. From the dispersion analysis, TF generally displays higher phase velocity and attenuation errors due to its first-order temporal accuracy nature compared to second-order ETD and TA. Although both have second-order temporal accuracy, ETD has generally lower phase velocity and attenuation errors than TA. This may be attributed to its closer resemblance to the solution of first-order differential equation. Numerical FDTD simulations in 1-D and 3-D further confirm these findings.
Optimal error estimates and energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell's equations  [PDF]
Liping Gao,Bo Zhang
Mathematics , 2011,
Abstract: This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell equations. Precisely, for the case with a perfectly electric conducting (PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete $H^1$-norm for the ADI-FDTD scheme and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero then the discrete $L^2$-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. A key ingredient is two new discrete energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell equations introduced in this paper. Furthermore, we prove that, in addition to two known discrete energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete energy identities which are second-order in time perturbations of the two new energy conservation laws. This means that the ADI-FDTD scheme is unconditionally stable under the four discrete energy norms. Experimental results are presented which confirm the theoretical results.
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