Analysis of the numerical dispersion of the 2Dalternating-direction implicit FDTD method

The numerical dispersion property of the two-dimensional alternating-direction implicit finite-difference time-domain (2D ADI FDTD) method is studied. First, we notice that the original 2D ADI FDTD method can be divided into two sub-ADI FDTD methods: either the x-directional 2D ADI FDTD method or the y-directional 2D ADI FDTD method; and secondly, the numerical dispersion relations are derived for both the ADI FDTD methods. Finally, the numerical dispersion errors caused by the two ADI FDTD methods are investigated. Numerical results indicate that the numerical dispersion error of the ADI FDTD methods depends highly on the selected time step and the shape and mesh resolution of the unit cell. It is also found that, to ensure the numerical dispersion error within certain accuracy, the maximum time steps allowed to be used in the two ADI FDTD methods are different and they can be numerically determined     

On the dispersion errors related to (FD)2TD type schemes

The dispersion errors associated with various frequency-dependent FDTD methods are considered herein. Particularly, we provide a rigorous error analysis of both direct integration and recursive type schemes for two media models: the one-pole Debye and the two-pole Lorentz. The error equations are cast in terms of a dispersion relation that shows explicitly the errors associated with numerically induced dispersion and dissipation. From the dispersion relation, plots are provided that typify the errors of each method. In general, all methods have about the same propagation characteristics; the differences, however, are seen in the attenuation plots. To validate the claims herein, data obtained from FDTD scattering simulations (both 1-D and 3-D geometries) are also given     

A three-dimensional angle-optimized finite-difference time-domain algorithm

We present a three-dimensional finite-difference time-domain (FDTD) algorithm to minimize the numerical dispersion error at preassigned angles. Filtering schemes are used to further optimize its frequency response for broad-band simulations. A stability analysis of the resulting FDTD algorithm is also provided. Numerical results show that the dispersion error around any preassigned angle can be reduced significantly in a broad range of frequencies with small computational overhead.     

A Highly Accurate FDTD Model for Simulating Lorentz Dielectric Dispersion

A highly accurate and numerically stable model of Lorentz dielectric dispersion for the finite-difference time-domain (FDTD) method is presented. The coefficients of the proposed model are optimally derived based on the Maclaurin series expansion (MSE) method and it is shown that the model is much better than the other four reported models in implementing the Lorentz dielectric dispersion with error of relative permittivity several orders lower. The model's stability and performance are also analyzed when it is incorporated into the practical second- and fourth-order accurate FDTD algorithms for an exemplified Lorentz medium. Interestingly, we find that all the mentioned models show nearly the same performance in the second-order algorithm due to its large intrinsic numerical dispersion and the superiority of the proposed MSE model begins to be manifested in the higher-order, say, fourth-order FDTD algorithms as implied by the governing numerical dispersion equations.      

Relative accuracy of several finite-difference time-domain methodsin two and three dimensions

A comparison of the accuracy of several orthogonal-grid finite-difference-time-domain (FDTD) schemes is made in both two and three-dimensions. The relative accuracy is determined from the dispersion error associated with each algorithm and the number of floating-point operations required to obtain a desired accuracy level. In general, in both 2-D and 3-D, fourth-order algorithms are more efficient than second-order schemes in terms of minimizing the number of computations for a given accuracy level. In 2-D, a second-order approach proposed by Z. Chen et al. (1991) is much more accurate than the scheme of K.S. Yee (1966) for a given amount of computation, and can be as efficient as fourth-order algorithms. In 3-D, Yee's algorithm is slightly more efficient than the approach of Chen et al. in terms of operations, but much more efficient in terms of memory requirements     

A general method for FDTD modeling of wave propagation in arbitraryfrequency-dispersive media

A general formulation is presented for finite-difference time-domain (FDTD) modeling of wave propagation in arbitrary frequency-dispersive media. Two algorithmic approaches are outlined for incorporating dispersion into the FDTD time-stepping equations. The first employs a frequency-dependent complex permittivity (denoted Form-1), and the second employs a frequency-dependent complex conductivity (denoted Form-2). A Pade representation is used in Z-transform space to represent the frequency-dependent permittivity (Form-1) or conductivity (Form-2). This is a generalization over several previous methods employing either Debye, Lorentz, or Drude models. The coefficients of the Pade model may be obtained through an optimization process, leading directly to a finite-difference representation of the dispersion relation, without introducing discretization error. Stability criteria for the dispersive FDTD algorithms are given. We show that several previously developed dispersive FDTD algorithms can be cast as special cases of our more general framework. Simulation results are presented for a one-dimensional (1-D) air/muscle example considered previously in the literature and a three-dimensional (3-D) radiation problem in dispersive, lossy soil using measured soil data     

FDTD Schemes With Minimal Numerical Dispersion

A novel formulation of hybrid finite-difference time-domain (FDTD) methods is presented. Significant reduction of numerical dispersion is achieved by the proposed FDTD methods that combine the second-order and higher-order finite-differences. Also, the proposed FDTD methods exhibit significantly higher solution accuracy than the accuracy of standard FDTD schemes as a result of partial mutual cancellation of numerical errors provided by the developed FDTD update procedure. The residual numerical error of the phase velocity remains low even for sampling of a few points per wavelength. Also, the FDTD schemes based on the proposed approach are faster and more accurate than the corresponding purely higher-order FDTD schemes with the same mesh. Test examples are provided for validation purposes.      

Development of higher order FDTD schemes with controllable dispersion error

A new methodology that facilitates the control of the inherent dispersion error in the case of higher order finite-difference time-domain (FDTD) schemes is presented in this paper. The basic idea is to define suitable algebraic expressions that reflect numerical inaccuracies reliably. Then, finite-difference operators are determined via the minimization of the error estimators at selected frequencies. In order to apply this procedure, an error expansion in terms of cylindrical harmonic functions is performed, which also enables accuracy enhancement for all propagation angles. The design process produces a set of two-dimensional (2-D) FDTD algorithms with optimized frequency response. Contrary to conventional methodologies, the proposed techniques adjust their reliability range according to the requirements of the examined problem and can be, therefore, more efficient in computationally demanding simulations.     

有耗介质中时域精细积分方法的数值色散分析

分析了时间步长、空间步长、电导率和电磁波传播方向对时域精细积分(PITD)方法的数值损耗和数值色散的影响。结果表明:PITD的数值损耗大于电磁波的真实损耗,其数值波速可以大于电磁波的真实波速。PITD的数值损耗和数值色散都基本上不受时间步长的影响。随着空间步长的减小,PITD的数值损耗和数值色散的误差都逐步减小。当电导率较小时,PITD的数值损耗和数值色散的误差比时域有限差分(FDTD)方法的大。但当电导率较大时,PITD的数值波速却比FDTD的数值波速更加接近于电磁波的真实波速。PITD的数值损耗和数值色散的各向异性在三维情况下的值要大于其在二维情况下的值。数值算例表明:对良导体而言,PITD比FDTD拥有更高的计算精度和更快的计算速度。     

FDTD-Modelling of Dispersive Nonlinear Ring Resonators: Accuracy Studies and Experiments

The accuracy of nonlinear finite-difference time-domain (FDTD) methods is investigated by modeling nonlinear optical interaction in a ring resonator. We have developed a parallelized 3-D FDTD algorithm which incorporates material dispersion, chi⁽³⁾-nonlinearities and stair-casing error correction. The results of this implementation are compared with experiments, and intrinsic errors of the FDTD algorithm are separated from geometrical uncertainties arising from the fabrication tolerances of the device. A series of progressively less complex FDTD models is investigated, omitting material dispersion, abandoning the stair-casing error correction, and approximating the structure by a 2-D effective index model. We compare the results of the different algorithms and give guidelines as to which degree of complexity is needed in order to obtain reliable simulation results in the linear and the nonlinear regime. In both cases, incorporating stair-casing error correction and material dispersion into a 2-D effective index model turns out to be computationally much cheaper and more effective than performing a fully three-dimensional simulation without these features     

MRTD: new time-domain schemes based on multiresolution analysis

The application of multiresolution analysis to Maxwell's equations results in new multiresolution time-domain (MRTD) schemes with unparalleled inherent properties. In particular, the approach allows the development of MRTD schemes which are based on scaling functions only or on a combination of scaling functions and wavelets leading to a variable mesh grading. The dispersion of the MRTD schemes compared to the conventional Yee finite-difference time-domain (FDTD) scheme shows an excellent capability to approximate the exact solution with negligible error for sampling rates approaching the Nyquist limit. Simple microwave structures including dielectric materials are analyzed in order to illustrate the application of the MRTD schemes and to demonstrate the advantages over Yee's FDTD scheme with respect to memory requirements and execution time     

辛算法的稳定性及数值色散性分析

引入一种新的数值计算方法 —辛算法求解Maxwell方程,即在时间上用不同阶数的辛差分格式离散,空间分别采用二阶及四阶精度的差分格式离散,建立了求解二维Maxwell方程的各阶辛算法,探讨了各阶辛算法的稳定性及数值色散性.通过理论上的分析及数值计算表明,在空间采用相同的二阶精度的中心差分离散格式时,一阶、二阶辛算法(T1S2、T2S2) 的稳定性及数值色散性与时域有限差分(FDTD)法一致,高阶辛算法的稳定性与FDTD法相当;四阶辛算法结合四阶精度的空间差分格式(T4S4) 较FDTD法具有更为优越的数值色散性.对二维TMz波的数值计算结果表明,高阶辛算法较FDTD法有着更大的计算优势.     

一种新型的高阶时域有限差分方法

相比于传统高阶时域有限差分算法(FDTD)而言,该文提出了一种改进的高阶FDTD的优化方法,该算法基于安培环路定律,通过计算机技术寻找到一组最优的系数使得FDTD方法的全局色散误差达到最小,通过不同分辨率下的点源辐射模拟证明了该方法在较低分辨率的情况下仍然具有极低的相位误差,对于解决电大尺寸结构建模中的数值色散等问题提供了有效的解决方案。     

Rigorous analysis of the influence of the aspect ratio of Yee's unit cell on the numerical dispersion property of the 2-D and 3-D FDTD methods

In this paper, the influence of the aspect ratio of Yee's unit cell on the numerical dispersion errors [in terms of the physical phase-velocity error (PVE) and the velocity-anisotropy error (VAE)] of two-dimensional (2-D) finite-difference time-domain (FDTD) and three-dimensional (3-D) FDTD methods is comprehensively investigated. Numerical results reveal that, for a fixed mesh resolution, the physical PVE and the VAE of both the 2-D and 3-D FDTD methods converge to certain limits for higher aspect ratio. Most importantly, it is found for the first time that for the 2-D and 3-D cases the converged dispersion errors (i.e., the limits) are, respectively, about 2.0 and 1.5 times of the corresponding square and cubic unit cells; and the validity of the above theoretical prediction is verified through numerical tests. The investigation carried out in this paper certainly confirms, from the numerical dispersion point of view, that very accurate numerical results can still be obtained even when the aspect ratio of the cells is higher. Consequently, it gives design engineers more freedom and confidence to use the FDTD methods, especially when the aspect ratio of the cells has to be greatly adjusted due to the special requirement of structures under study.     

Analysis and compensation of numerical dispersion in the FDTD method for layered, anisotropic media

In this work, we investigate the effects of numerical dispersion in the finite-difference time-domain (FDTD) algorithm for layered, anisotropic media. We first derive numerical dispersion relations for diagonally anisotropic media (corresponding to an FDTD reference frame coinciding with the principal axes of a biaxial media). In addition, we incorporate the discretization effects on the reflection and transmission coefficients in layered media. We then apply this analysis to minimize the numerical dispersion error of Huygens' plane-wave sources in layered, uniaxial media. For usual discretization sizes, a typical reduction of the scattered field error on the order of 30 dB is demonstrated.     

Optimization of subgridding schemes for FDTD

A procedure to optimize the coupling coefficients between fine and coarse mesh regions for two-dimensional (2-D) finite-difference time-domain (FDTD) subgridding algorithms is introduced. The coefficients are optimized with respect to different angles and expanded in a form suitable for FDTD computation     

Generalized dispersion analysis and spurious modes of 2-D and 3-DTLM formulations

The general dispersion relations are derived for the 2-D TLM shunt and series meshes and the 3-D TLM expanded and condensed node meshes. Implicit in the resulting dispersion relations are both their physical and spurious modal solutions. It is demonstrated that of the four schemes, only the 3-D expanded node mesh is free of detrimental spurious solutions     

A FDTD-based modal PML

The finite-difference time-domain (FDTD) method is one of the most popular numerical methods for solving electromagnetic problems because of its algorithmic simplicity and flexibility. For an open waveguide structure, modal perfectly matched layer (PML) schemes have been developed as efficient absorbing terminations. However, since these PML schemes are not derived directly from the FDTD algorithm, they do not perform as well as the original three-dimensional (3-D) PMLs. In this letter, a FDTD-based one-dimensional modal PML is proposed. Because it is derived directly from the FDTD formulation, its numerical dispersion characteristics are very close to the original FDTD method. Relative differences between results obtained with the proposed method and the original 3-D PML are found to be less than -220dB, and the proposed modal PML is shown to perform at least the same as the original PML if not better.     

A Finite Volumes-Based 3-D Low Dispersion FDTD Algorithm

An algorithm extension to three dimensions is developed and presented for the highly phase-coherent modified second-order in time, fourth-order in space (or M24) finite-difference time-domain (FDTD) algorithm. A finite-volumes approach in conjunction with Yee's standard FDTD lattice is used for algorithm development. The corresponding dispersion relation is also developed, analyzed and compared to both the standard second-order and fourth-order FDTD algorithms as well as to two closely related high-order phase-coherent algorithms. Wideband algorithm attributes are also presented as well as sets of ready to use optimized algorithm coefficients.     

Research on the Radiation Characteristics of Patched Leaky Coaxial Cable by FDTD Method and Mode Expansion Method

Patched leaky coaxial cable (PLCX) is proposed as an alternative to the conventional leaky cable for wireless links in a complex environment. It is expected to have the capability of adjusting the coupling between the cable and the environment and give smoother electric field coverage. In this paper, the radiation characteristics of the PLCX with general inclined patches are studied by a hybrid method that involves the finite-difference time-domain (FDTD) method for the near-field computation and the mode expansion method for the transformation of near field to far field. In the method, the space around the patched leaky cable is divided into two regions by an artificial closed cylindrical surface that is incorporated with the FDTD lattice surface when implementing the FDTD iteration. The field distribution on the artificial surface is obtained after the implementation of the FDTD method. Meanwhile, the field outside the artificial boundary is expanded in terms of the Floquet modes with coefficients to be determined. By matching the field expressed by modes and the field obtained from the FDTD method at the artificial boundary, a matrix equation with unknown coefficients is obtained. Solving this matrix equation, the expansion coefficients are known, and the field outside the artificial boundary is ready to be obtained.