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NUMERICAL STABILITY AND NUMERICAL DISPERSION OF CONFORMAL MAPPING FDTD ALGORITHM
保角变换FDTD算法的数值稳定性与数值色散

Zhou Xiaojun,Yu Zhiyuan,Lin Weigan,
周晓军
,喻志远,林为干

电子与信息学报 , 2000,
Abstract: This paper proposes a new algorithm based on conformal mapping and FDTD method, and derives the numerical stability and numerical dispersion equations of conformal mapping FDTD algorithm. As an example, the relative errors of numerical wavelengths for TE modes in a circular waveguide in different cell number are calculated. The errors for different propagation constants and for different radius semicircle electric wall approaches to singularity at origin are analyzed. By selecting cell number appropriately, high accuracy can be obtained.
An Efficient Method to Reduce the Numerical Dispersion in the HIE-FDTD Scheme  [PDF]
Juan Chen, Anxue Zhang
Wireless Engineering and Technology (WET) , 2011, DOI: 10.4236/wet.2011.21005
Abstract: A parameter optimized approach for reducing the numerical dispersion of the 3-D hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) is presented in this letter. By adding a parameter into the HIE-FDTD formulas, the error of the numerical phase velocity can be controlled, causing the numerical dispersion to decrease significantly. The numerical stability and dispersion relation are presented analytically, and numerical experiments are given to substantiate the proposed method.
Analysis of the Numerical Dispersion of Higher Order ADI-FDTD
高阶ADI-FDTD算法的数值色散分析

Xu Li-jun,Yuan Nai-chang,
徐利军
,袁乃昌

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.
Two Efficient Unconditionally-Stable Four-Stages Split-Step FDTD Methods with Low Numerical Dispersion
Yong-Dan Kong;Qing-Xin Chu;Rong-Lin Li
PIER B , 2013, DOI: 10.2528/PIERB12103011
Abstract: Two efficient unconditionally-stable four-stages split-step (SS) finite-difference time-domain (FDTD) methods based on controlling parameters are presented, which provide low numerical dispersion. Firstly, in the first proposed method, the Maxwell's matrix is split into four sub-matrices. Simultaneously, two controlling parameters are introduced to decrease the numerical dispersion error. Accordingly, the time step is divided into four sub-steps. The second proposed method is obtained by adjusting the sequence of the sub-matrices deduced in the first method. Secondly, the theoretical proofs of the unconditional stability and dispersion relations of the proposed methods are given. Furthermore, the processes of obtaining the controlling parameters for the proposed methods are shown. Thirdly, the dispersion characteristics of the proposed methods are also investigated, and numerical dispersion errors of the proposed methods can be decreased significantly. Finally, to substantiate the efficiency of the proposed methods, numerical experiments are presented.
Numerical Dispersion and Impedance Analysis for 3D Perfectly Matched Layers Used for Truncation of the FDTD Computations
Weiliang Yuan;Er Ping Li
PIER , 2004, DOI: 10.2528/PIER03121002
Abstract: The 3D Berenger's and uniaxial perfectly matched layers used for the truncation of the FDTD computations are theoretically investigated respectively in the discrete space, including numerical dispersion and impedance characteristics. Numerical dispersion for both PMLs is different from that of the FDTD equations in the normal medium due to the introduction of loss. The impedance in 3D homogeneous Berenger's PML medium is the same as that in the truncated normal medium even in the discrete space, however, the impedance in 3D homogenous UPML medium is different, but the discrepancy smoothly changes as the loss in the UPML medium slowly change. Those insights acquired can help to understand why both 3D PMLs can absorb the outgoing wave with arbitrary incidence, polarization, and frequency, but with different efficiency.
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.
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.
Study for the Numerical Properties of the Higher-Order LOD-FDTD Methods
高阶LOD-FDTD方法的数值特性研究

Liu Guo-sheng,Zhang Guo-ji,
刘国胜
,张国基

电子与信息学报 , 2010,
Abstract: In this paper, the numerical properties of higher-order Locally One Dimensionally Finite-Difference Time-Domain (LOD-FDTD) are investigated, i.e. stability, numerical dispersion, and convergence. The universal formulas of the amplitude factor and the numerical dispersion relationship are derived for 3D varying-order LOD-FDTD, and their unconditional stability is analytically proved. Based on this universal formula, the numerical convergence of this class of methods is discussed, and the convergence condition is presented for the first time. Numerical results in calculating the resonant frequency show that, higher-order methods can achieve better performance while not dramatically increasing computational time.
Conformal mapping methods for interfacial dynamics  [PDF]
Martin Z. Bazant,Darren Crowdy
Physics , 2004, DOI: 10.1007/978-1-4020-3286-8_71
Abstract: The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.
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.
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