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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.
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.
Unconditionally Stable Leapfrog Adi-FDTD Method for Lossy Media
Theng Huat Gan;Eng Leong Tan
PIER M , 2012, DOI: 10.2528/PIERM12090307
Abstract: This paper presents an unconditionally stable threedimensional (3-D) leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method for lossy media. Conductivity terms of lossy media are incorporated into the leapfrog ADI-FDTD method in an analogous manner as the conventional explicit FDTD method since the leapfrog ADI-FDTD method is a perturbation of the conventional explicit FDTD method. Implementation of the leapfrog ADI-FDTD method for lossy media with special consideration for boundary condition is provided. Numerical results and examples are presented to validate the formulation.
Analysis of the Numerical Dispersion of Higher Order ADI-FDTD

Xu Li-jun,Yuan Nai-chang,

电子与信息学报 , 2005,
Abstract: In this paper, a new higher order Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) formulation in particular, a second-order-in-time, fourth-order-in-space AD-FDTD method is presented for the first time. At the same time ,the unconditional stability of the higher order ADI-FDTD formulation is analytically proved. By analysis of the amplification factors, the numerical dispersion relation is derived. In addition, the numerical dispersion errors are investigated. Finally numerical results indicate that the higher order ADI-FDTD has s better accuracy compared to the ADI-FDTD method.
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.
A Hybrid Method Based on ADI-FDTD and Its Celerity Algorithm

Zhang WeiJun,Yuan NaiChang,Li Yi,Zheng QiuRong,

电子与信息学报 , 2005,
Abstract: The paper presents a hybrid method based on FDTD and ADI-FDTD, introduces a method of weighted average to reduce the reflection caused by two different methods on the boundary, and presents a celerity method for the linear, lossless and isotropic medium. The simulation shows that the method is feasible.
The Adi-FDTD Method Including Lumped Networks Using Piecewise Linear Recursive Convolution Technique
Fen Xia;Qing-Xin Chu;Yong-Dan Kong;Zhi-Yong Kang
PIER M , 2013,
Abstract: The lumped network alternating direction implicit finite difference time domain (LN-ADI-FDTD) technique is proposed as an extension of the conventional ADI-FDTD method in this paper, which allows the lumped networks to be inserted into some ADI-FDTD cells. Based on the piecewise linear recursive convolution (PLRC) technique, the current expression of the loaded place can be obtained. Then, substituting the expression into the ADI-FDTD formulas, the difference equations including an arbitrary linear network are derived. For the sake of showing the validity of the proposed scheme, lumped networks are placed on the microstrip and the voltage across the road is computed by the lumped network finite difference time domain (LN-FDTD) method and LN-ADI-FDTD method, respectively. Moreover, the results are compared with those of obtained by using the circuital simulator ADS. The agreement among all the simulated results is achieved, and the extended ADI-FDTD method has been shown to overcome the Courant-Friedrichs-Lewy (CFL) condition.
A Hybrid Implicit-Explicit Spectral FDTD Scheme for Oblique Incidence Problems on Periodic Structures
Yunfei Mao;Bin Chen;Hao-Quan Liu;Jing-Long Xia;Ji-Zhen Tang
PIER , 2012, DOI: 10.2528/PIER12032306
Abstract: This paper combines a hybrid implicit-explicit (HIE) method with spectral finite-difference time-domain (SFDTD) method for solving periodic structures at oblique incidence, resulting in a HIE-SFDTD method. The new method has the advantages of both HIE-FDTD and SFDTD methods, not only making the stability condition weaker, but also solving the oblique incident wave on periodic structures. Because the stability condition is determined only by two space discretizations in this method, it is extremely useful for periodic problems with very fine structures in one direction. The method replaces the conventional single-angle incident wave with a constant transverse wave-number (CTW) wave, so the fields have no delay in the transverse plane, as a result, the periodic boundary condition (PBC) can be implemented easily for both normal and oblique incident waves. Compared with the ADI-SFDTD method it only needs to solve two untridiagonal matrices when the PBC is applied to, other four equations can be updated directly, while four untridiagonal matrices, two tridiagonal matrices, and six explicit equations should be solved in the ADI-SFDTD method. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed algorithm. Results show the new algorithm has better accuracy and higher efficiency than that of the ADI-SFDTD method, especially for large time step sizes. The CPU running time for this method can be reduced to about 45% of the ADI-SFDTD method.
Application of Artificial Anisotropy in 3-D ADI-FDTD Method

Zhang Yan,,Shan-wei,
张 岩

电子与信息学报 , 2006,
Abstract: Attention is focused on a new method to reduce the numerical dispersion of the 3-D Alternating-Direction Implicit Finite-Difference Time-Domain(ADI-FDTD) method through artificial anisotropy. As the wave propagation can be speeded up by introducing proper anisotropy parameters into the 3-D ADI-FDTD method, the numerical dispersion can be reduced and the accuracy can be improved significantly. First, the numerical formulations of the 3-D ADI-FDTD method are modified. Secondly, the new numerical dispersion relation is derived. And consequently the relative permittivity tensor of artificial anisotropy can be obtained. In order to demonstrate the accuracy and efficiency of this new method, a hollow waveguide and a waveguide with discontinuous structure are simulated as examples. In addition the reduction of numerical dispersion is investigated as a function of the relative permittivity tensor of artificial anisotropy. Furthermore, the numerical results and the computational requirements of the proposed method are compared with those of the conventional 3-D ADI-FDTD method. It is found that this new method is accurate and efficient.
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