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Reduction of Numerical Dispersion of the Six-Stages Split-Step Unconditionally-Stable FDTD Method with Controlling Parameters
Yong-Dan Kong;Qing-Xin Chu
PIER , 2012, DOI: 10.2528/PIER11082512
Abstract: A new approach to reduce the numerical dispersion of the six-stages split-step unconditionally-stable finite-difference time-domain (FDTD) method is presented, which is based on the split-step scheme and Crank-Nicolson scheme. Firstly, based on the matrix elements related to spatial derivatives along the x, y, and z coordinate directions, the matrix derived from the classical Maxwell's equations is split into six sub-matrices. Simultaneously, three controlling parameters are introduced to decrease the numerical dispersion error. Accordingly, the time step is divided into six sub-steps. Secondly, the analysis shows that the proposed method is unconditionally stable. Moreover, the dispersion relation of the proposed method is carried out. Thirdly, the processes of determination of the controlling parameters are shown. Furthermore, the dispersion characteristics of the proposed method are also investigated, and the maximum dispersion error of the proposed method can be decreased significantly. Finally, numerical experiments are presented to substantiate the efficiency of the proposed method.
Applications of a Three-Dimensional FDTD Method with Weakly Conditional Stability to the Analysis of Microstrip Filters with Fine Scale Structures
Jing Lan;Yang Yang;Jing Yi Dai
PIER Letters , 2011, DOI: 10.2528/PIERL11082213
Abstract: In three-dimensional space, the hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is weakly conditionally stable, only determined by two space-discretizations, which is very useful for problems with fine structures in one direction. Its numerical dispersion errors with nonuniform cells are discussed and compared in this paper. To enlarge the applicable field of the HIE-FDTD method to open space, the absorbing boundary conditions (ABCs) for this method are also introduced and applied. Two microstrip filters with fine scale structures in one direction are solved by the HIE-FDTD method. Conventional FDTD method and alternating-direction implicit FDTD (ADI-FDTD) method are also used for comparing. Results analyzed by the HIE-FDTD method agree well with those from conventional FDTD, and the required central process unit (CPU) time is much less than that of the ADI-FDTD method.
共形HIE-FDTD方法的稳定性分析及改进  [PDF]
重庆邮电大学学报(自然科学版) , 2013,
Abstract: 提出一种金属共形混合隐显式时域有限差分方法(conformalhybridimplicitexplicitfinitedifferencetimedomain,CHIE-FDTD),与传统FDTD方法相比,该方法用于计算不能与网格共形的金属目标时,既能克服阶梯近似法带来的误差和引起的虚拟表面波,又能提高计算效率。但由于Dey-Mittre金属共形方法本身要求减小柯朗弗里德里希斯列维(Courant-Friedrichs-Lewy,CFL)条件来增加稳定性,将其引入HIE-FDTD会导致时间步长的减小或使得仿真波形发散提前(降低算法的稳定性)。针对该问题开展了研究,给出了改进方法,并进行了数值验证。
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.
Interfacial Numerical Dispersion and New Conformal FDTD Method  [PDF]
Axman Fisher
Physics , 2011,
Abstract: This article shows the interfacial relation in electrodynamics shall be corrected in discrete grid form which can be seen as certain numerical dispersion beyond the usual bulk type. Furthermore we construct a lossy conductor model to illustrate how to simulate more general materials other than traditional PEC or simple dielectrics, by a new conformal FDTD method which main considers the effects of penetrative depth and the distribution of free bulk electric charge and current.
FDTD Analysis of the Dispersion Characteristics of the Metal PBG Structures
Ashutosh;Pradip Kumar Jain
PIER B , 2012, DOI: 10.2528/PIERB11120601
Abstract: Two dimensional metallic photonic band gap (PBG) structures, which have higher power handling capability, have been analyzed for their dispersion characteristics. The analysis has been performed using finite difference time domain (FDTD) method based on the regular orthogonal Yee's cell. A simplified unit cell of triangular lattice PBG structure has been considered for the and modes of propagation. The EM field equations in the standard central-difference form have been taken in FDTD method. Bloch's periodic boundary conditions have been used by translating the boundary conditions along the direction of periodicity. For the source excitation, a wideband Gaussian pulse has been used to excite the possible modes in the computational domain. Fourier transform of the probed temporal fields has been calculated which provides the frequency spectrum for a set of wave vectors. The determination of eigenfrequencies from the peaks location in the frequency spectrum has been described. This yields the dispersion diagram which describes the stop and pass bands characteristics. Effort has been made to describe the estimation of defect bands introduced in the PBG structures. Further, the present orthogonal FDTD results obtained have been compared with those obtained by a more involved non-orthogonal FDTD method. The universal global band gap diagrams for the considered metal PBG structure have been obtained by varying the ratio of rod radius to lattice constant for both polarizations and are found identical with those obtained by other reported methods. Convergence of the analysis has been studied to establish the reliability of the method. Usefulness of these plots in designing the devices using 2-D metal PBG structure has also been illustrated.
A Novel RC-FDTD Algorithm for the Drude Dispersion Analysis
Antonino Cala' Lesina;Alessandro Vaccari;Alessandro Bozzoli
PIER M , 2012, DOI: 10.2528/PIERM12041904
Abstract: One of the main techniques for the Finite-Difference Time-Domain (FDTD) analysis of dispersive media is the Recursive Convolution (RC) method. The idea here proposed for calculating the updating FDTD equation is based on the Laplace transform and is applied to the Drude dispersion case. A modified RC-FDTD algorithm is then deduced. We test our algorithm by simulating gold and silver nanospheres exposed to an optical plane wave and comparing the results with the analytical solution. The modified algorithm guarantees a better overall accuracy of the solution, in particular at the plasmonic resonance frequencies.
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.
An Unconditional Stable 1D-FDTD Method for Modeling Transmission Lines Based on Precise Split-Step Scheme
Wei Wang;Pei-Guo Liu;Yu-Jian Qin
PIER , 2013, DOI: 10.2528/PIER12103007
Abstract: his paper presented a novel unconditional stable FDTD (US-FDTD) algorithm for solving the transient response of uniform or nonuniform multiconductor transmission line with arbitrary coupling status. Analytical proof of unconditional stability and detailed analysis of numerical dispersion are presented. The precise split-time-step scheme has been introduced to eliminate the restriction of the Courant-Friedrich-Levy (CFL) condition. Compared to the conventional USFDTD methods, the proposed approach generally achieves lower phase velocity error for coarse temporal resolution. So larger time scales can be chosen for the transient simulation to achieve accurate results efficiently. Several examples of coupled uniform and nonuniform lines are presented to demonstrate the accuracy, stability, and efficiency of the proposed model.
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