ADI-FDTD Method With Fourth Order Accuracy in Time

This letter presents an unconditionally stable alternating direction implicit finite-difference time-domain (ADI-FDTD) method with fourth order accuracy in time. Analytical proof of unconditional stability and detailed analysis of numerical dispersion are presented. Compared to second order ADI-FDTD and six-steps SS-FDTD, the fourth order ADI-FDTD generally achieves lower phase velocity error for sufficiently fine mesh. Using finer mesh gridding also reduces the phase velocity error floor, which dictates the accuracy limit due to spatial discretization errors when the time step size is reduced further.     

一种新型全域时域有限差分法

提出一种基于半导体器件漂移扩散模型并结合交替方向隐式时域有限差分(ADI-FDTD)法的新型全域FDTD法.该方法时间步长的选择取决于数值精度而非稳定性,克服了传统全域FDTD法为了保持数值稳定性,时间步长的选取受限于Courant稳定性条件的问题.该方法的复杂度稍微增加,但是通过增加时间步长,减少计算时间、提高计算速度.该方法的主要特点是模拟一个简单的二极管分布开关电路.     

新型无条件稳定SSCN-FDTD算法

提出了一种新型的基于split-step方案和Crank-Nicolson方案的时域有限差分法(finite-difference timedomain method FDTD),并且证明了此种算法的无条件稳定性.所提出的算法采用新的矩阵分解形式,沿着x、y、z三个方向进行分解,将三维问题转化为一维问题,与alternating direction implicit(ADI)-FDTD算法、split-step(SS)-FDTD(1,2)算法和SS-FDTD(2,2)算法相比,减少了计算复杂度,提高了计算效率;同时所提出的算法具有二阶时间精度和二阶空间精度.新型算法的推导程序比基于指数因子分解的无条件FDTD算法更简单.将新型算法用于计算谐振腔结构,在计算相对误差一致的情况下,计算时间比ADI-FDTD算法节省约31%,比SS-FDTD(1,2)算法节省约13.5%.     

Conformal Wavelet Finite-difference Time-domain Algorithm for Analysis of Milimeter Wave Attenuation on Coplanar Waveguide

Addressed is the calculation of millimeter wave attenuation on coplanar waveguide（CPW）. A novel conformal wavelet finite-difference time-domain（CWFDTD） algorithm is proposed with emphasis on its application in calculation of millimeter wave attenuation on CPW, which is the combination of conformal algorithm dealing with the deformed cell with Wavelet-FDTD using multi-resolution analysis（MRA）. Derived is the difference formulation for multi-resolution time domain（MRTD） based on Daubechies wavelets, and also given is the stability conditions for wavelet-FDTD algorithm. To validate its accuracy and efficiency, this novel method is applied to calculate the millimeter wave attenuation on lithium niobate CPW. Numerical results demonstrate that this new CWFDTD algorithm has the same accuracy with the conformal finite-difference timedomain（CFDTD） and conformal finite-difference time-domain based on alternating-direction implicit method （ADI-CFDTD）, but saves computational time and computer memory.     

An Arbitrary-Order LOD-FDTD Method and its Stability and Numerical Dispersion

An arbitrary-order unconditionally stable three-dimensional (3-D) locally-one- dimensional finite-difference time-method (FDTD) (LOD-FDTD) method is proposed. Theoretical proof and numerical verification of the unconditional stability are shown and numerical dispersion is derived analytically. Effects of discretization parameters on the numerical dispersion errors are studied comprehensively. It is found that the second-order LOD-FDTD has the same level of numerical dispersion error as that of the unconditionally stable alternating direction implicit finite-difference time-domain (ADI-FDTD) method and other LOD-FDTD methods but with higher computational efficiency. To reduce the dispersion errors, either a higher-order LOD-FDTD method or a denser grid can be applied, but the choice has to be carefully made in order to achieve best trade-off between the accuracy and computational efficiency. The work presented in this paper lays the foundations and guidelines for practical uses of the LOD method including the potential mixed-order LOD-FDTD methods.      

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     

Reducing the phase error for finite-difference methods withoutincreasing the order

The phase error in finite-difference (FD) methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy. Greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods. Both these techniques require more memory and longer computation times. In this paper, new modified methods are presented which are optimized to problems of electromagnetics. Simple methods are presented that reduce numerical phase error without additional processing time or memory requirements. Furthermore, these methods are applied to both the Helmholtz equation in the frequency domain and the finite-difference time-domain (FDTD) method. Both analytical and numerical results are presented to demonstrate the accuracy of these new methods     

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.      

Analysis of Two Alternative ADI-FDTD Formulations for Transverse-Electric Waves in Lossy Materials

A numerical dispersion analysis of the alternating-direction implicit finite-difference time-domain method for transverse-electric waves in lossy materials is presented. Two different finite-difference approximations for the conduction terms are considered: the double-average and the synchronized schemes. The numerical dispersion relation is derived in a closed form and validated through numerical simulations. This study shows that, despite its popularity, the accuracy of the double-average scheme is sensitive to how well the relaxation-time constant of the material is resolved by the time step. Poor resolutions lead to unacceptably large numerical errors. On the other hand, for good conductors, the synchronized scheme allows stability factors as large as 100 to be used without deteriorating the accuracy significantly.      

Higher-order FDTD(2,4) scheme for accurate simulations in lossy dielectrics

A finite-difference time-domain (FDTD)(2,4) scheme with second-order accuracy in time and fourth-order accuracy in space for the precise solution of Maxwell's equations in lossy dielectrics is presented. Compared with the ordinary FDTD method the novel technique reduces lattice reflection errors, increases the overall accuracy and provides significant computational savings. Numerical results for a waveguide problem indicate the efficiency and robustness of the proposed formulation.     

Optimized ADI-FDTD analysis of circularly polarized microstrip and dielectric resonator antennas

A dispersion-optimized alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is presented in this letter for the accurate modeling of complex electromagnetic structures. The analysis focuses on circularly polarized slot-coupled microstrip and dielectric resonator antennas. Introducing a class of three-dimensional spatial/temporal operators, the novel algorithm yields robust curvilinear grids which efficiently treat fine details and interfaces. Hence, the serious dispersion errors of the usual schemes, as time-steps surpass the Courant limit, are greatly reduced, enabling fast broad-band simulations. Numerical validation confirms the above merits via various realistic structures.     

Dispersion analysis of the three-dimensional complex envelope ADI-FDTD method

In this paper, numerical dispersion properties of the three-dimensional complex envelope (CE) alternate-direction implicit finite-difference time-domain (ADI-FDTD) method are studied. The variations of dispersion errors with propagation direction, ratio of carrier to envelope frequencies, and spatial and temporal steps are presented. It is found that the CE ADI-FDTD scheme have much better accuracy and efficiency over the ADI-FDTD, especially with a higher ratio of carrier to envelope frequencies. Therefore, the CE ADI-FDTD is recommended for use in efficient narrow bandwidth electromagnetic modeling.     

Second-order split-step envelope PML algorithm for 2D FDTD simulations

Unconditionally stable second-order split-step (SS) envelope perfectly matched layer (PML) formulations are presented for truncating finite-difference time-domain (FDTD) grids. The proposed method is based on the second-order SS-FDTD algorithm. Numerical examples carried out in two-dimensional domains are included to show the validity of the proposed formulations.     

一种色散介质中的低误差ADI-FDTD 算法

交变方向隐式时域有限差分（ADI-FDTD）能够克服传统时域有限差分算法中稳定性条件对时间步长的限制,从而提高计算效率,但是在大步长时其误差较大。ER（低误差）-ADI-FDTD 方法通过补偿截断误差项,提高了计算精度,但是目前仅给出二维非色散条件下的形式。在ER-ADI-FDTD 的基础上,提出了一种色散介质中的低误差D-ER-ADI-FDTD 算法,推导出了完整的三维计算公式。最后通过计算和结果比较对算法进行检验。     

截断各向异性介质的修正NPML吸收边界条件

基于各向异性介质中的时域有限差分（Finite-Difference Time-Domain,FDTD）方法及近似完全匹配层（Nearly Perfect Match Layer,NPML）原理,提出一种截断各向异性介质的修正的NPML吸收边界条件.通过对Maxwell旋度方程中的空间偏导数进行坐标拉伸并结合空间插值方法,推导出易于在FDTD方法中实现的吸收边界公式.计算了电偶极子辐射场的反射误差,验证了这种吸收边界截断二维各向异性介质的有效性.三维算例中数值模拟了时谐场的相位分布,以及不同网格NPML吸收层随时间变化的反射误差.数值结果表明NPML吸收边界能有效吸收各向异性介质中的电磁波.     

Stability and Numerical Dispersion Analysis of a Fourth-Order Accurate FDTD Method

In order to obtain high-order accuracy, a fourth-order accurate finite difference time-domain (FDTD) method is presented by Kyu-Pyung Hwang. Unlike conventional FDTD methods, a staggered backward differentiation scheme instead of the leapfrog scheme is used to approximate the temporal partial differential operator. However, the high order of its characteristic equation derived by the Von Neumann method makes the analysis of its numerical dispersion and stability very difficult. In automatic control theory, there are two effective methods for the stability analysis, i.e., the Routh–Hurwitz test and the Jury test. The combination of the Von Neumann method with each of the two can strictly derive the stability condition, which only makes use of the coefficients of its characteristic equation without numerically solving it. The method of analysis in this paper is also applicable in the stability and numerical analysis of other high-order accurate FDTD methods.     

High-order accurate split-step FDTD method for solution of maxwell?s equations

The split-step finite difference time domain (SS-FDTD) method characterised by unconditional stability is becoming an important numerical method in computational electromagnetics. Proposed is a new high-order accurate unconditionally stable SS-FDTD method, which is derived from the exponential evolution operator. Compared with the conventional SS-FDTD method, the numerical dispersion of the new method is greatly reduced     

An efficient method to reduce the numerical dispersion in the ADI-FDTD

A new approach to reduce the numerical dispersion in the finite-difference time-domain (FDTD) method with alternating-direction implicit (ADI) is studied. By adding anisotropic parameters into the ADI-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 discussed in this paper. Numerical experiments are given to substantiate the proposed method.     

Generalized analysis of stability and numerical dispersion in thediscrete-convolution FDTD method

A simple technique is described for determining the stability and numerical dispersion of finite-difference time-domain (FDTD) calculations that are linear, second-order in space and time, and include dispersion by the discrete convolution method. The technique is applicable to anisotropic materials. Numerical examples demonstrate the accuracy of the technique for several anisotropic and/or dispersive materials     

An Unconditionally Stable Split-Step FDTD Method for Low Anisotropy

Split-step unconditionally stable finite-difference time-domain (FDTD) methods have higher dispersion and anisotropic errors for large stability factors. A new split-step method with four sub-steps is introduced and shown to have much lower anisotropy compared with the well known alternating direction implicit finite-difference time-domain (ADI-FDTD) and other known split step methods. Another important aspect of the new method is that for each space step value there is a stability factor value that the numerical propagation phase velocity is isotropic.