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.     

基于Taylor级数展开定理的高阶FDTD的色散分析

时域有限差分法(FDTD)是计算电磁领域中的一类非常重要的研究工具.而Taylor级数展开定理是构造差分格式的一种重要方法,例如Yee格式采用二阶Taylor格式,Fang格式采用四阶Taylor格式.本文借助于采样定理,详细分析了不同阶Taylor中心差分格式的谱特性以及计算误差,并将任意阶Taylor中心差分格式用于数值求解麦克斯韦方程中,严格导出了稳定性条件和数值色散关系的表达式,引入了新的误差定义来衡量算法的好坏.详细地研究了Courant数、网格分辨率CPW和网格长度比率等因素对于数值色散误差的影响,为基于Taylor差分格式的FDTD算法的研究提供了有用的参考.     

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。     

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。     

基于高阶时域有限差分算法的电磁波传播计算

利用FDTD(2,4)高阶时域有限差分(Finite-Difference Time-Domain,FDTD)算法并结合滑动窗口的思想,对电磁波传播特性进行了仿真计算.采用的高阶FDTD算法在空间上达到四阶精度,与二阶精度的传统FDTD算法相比,在相同每波长采样数的条件下,数值色散误差能得到进一步的减少.在源脉冲传播较长距离时,数值色散的减少使得时域下脉冲扩展现象得到改善,滑动子窗口仍然能包含着激励源脉冲的全部信息,从而可更加准确地计算长距离电波传播特性.另外,在相同的数值色散误差容限下,每波长采样数比传统二阶FDTD方法有所减少,从而节省存储空间,加快计算速度.     

三维最优时域有限差分方法

数值色散是时域有限差分方法(FDTD)中最主要的误差来源,导致数值相速成为频率和方向的函数。文中讨论了一种基于最优有限冲激滤波器设计方法的最优差分格式,从频率空间或者波数空间中实现对理想偏微分算子的逼近,构造一种新的具有低数值色散关系的最优时域有限差分方法。文中导出了其数值色散关系和进行了稳定性分析,并通过与常用的基于泰勒级数展开定理的高阶(2,4)时域有限差分法相比较,发现最优时域有限差分法的数值色散得到了极大的改善。最后通过一个数值例子来验证其有效性。     

色散媒质中ADI-FDTD

基于交替方向隐式(ADI)技术的时域有限差分(FDTD)法是一种非条件稳定的计算方法，该方法的时间步长不受Courant稳定条件限制，而由数值色散误差决定。与传统的FDTD相比，ADI-FDTD增大了时间步长，从而缩短了总的计算时间。本文采用递归卷积方法将ADI-FDTD推广应用于色散媒质，推导了二维情况下色散媒质中的ADI-FDTD迭代公式。应用推导公式计算了色散土壤中目标的散射，并与色散媒质FDTD结果对比，在大量减少计算时间的情况下，两者结果符合很好。     

二维无条件稳定时域有限差分方法

基于半隐式的Crank-Nicolson差分格式给出了一种无条件稳定时城有限差分方法。和传统FDTD法中采用的显式差分格式不同，对Maxwell方程组采用半隐式差分格式，在时间和空间上仍然是二阶精确的。但时间步长不再受稳定性条件的限制，只需考虑数值色散误差对其取值的制约。利用分裂场完全匹配层吸收边界截断计算空间，为保证PML空间的无条件稳定性，其方程也采用半隐式差分格式。数值结果表明相同条件下US-FDTD方法与传统FDTD方法的计算精度是相同的，而且在增大时间步长时US-FDTD方法是稳定的和收敛的。可以预见US-FDTD方法在模拟具有电小结构问题时具有实际意义。     

高阶时间空间差分近似对二维ADI-FDTD方法数值色散的影响

本文讨论交替方向隐式时域有限差分法(ADI—FDTD)的数值色散问题，分别对高阶时间空间差分近似，介绍了近似公式，并进行2阶、4阶、6阶、10阶差分数值色散误差的算例计算，对比表明4阶空间差分近似产生的色散误差较小。     

高阶LOD-FDTD方法的数值特性研究

该文分析并证明了高阶局部1维时域有限差分(LOD-FDTD)方法的数值特性,即:稳定性、数值色散及高阶收敛性。文中首次推导出3维各阶LOD-FDTD方法的增长因子和数值色散关系的一致形式,解析证明了这类方法均是无条件稳定的。基于此一致性关系,首次分析了这类方法的数值色散误差随阶数的收敛情况,并给出收敛性条件。在用此类方法计算谐振腔本征模频率的实验中,数值结果显示高阶方法可达到更优的计算精度,同时不显著增加计算时间。     

UPML媒质中的包络ADI-FDTD

由于交替方向隐式时域有限差分法(Alternating-Direction Implicit Finite-Difference Time Domain,ADI-FDTD)的数值色散会随着时间步长的增加而增加,文中讨论了单轴各向异性完全匹配层(uniaxial perfectly matched layer,UPML)媒质中包络交替方向隐式时域有限差分法(Envelope ADI-FDTD),推导了二维Envelope ADI-FDTD UPML的迭代公式,并提出一种新的离散方法。与ADI-FDTD UPML相比,改进后的Envelope ADI-FDTD UPML的时间步长可以取得更大,且能有效地修正相速误差,从而减少数值色散,提高计算精度。     

各向异性介质在三维ADI-FDTD中的应用

该文研究一种减小三维交替方向隐式时域有限差分法(ADI-FDTD)数值色散的新方法。通过在三维空间中合理添加各向异性介质,达到调整相速的目的,从而减小数值色散,使计算结果更加精确。首先对添加各向异性介质后的三维ADI-FDTD迭代公式进行变形,并得到新的数值色散关系,从而求解得到各向异性介质的相对介电常数。以空心波导和具有介质不连续性的波导作为数值算例,分析不同的各向异性介质和添加方法对计算精度的影响,并与传统ADI-FDTD得到的结果和计算资源占用情况进行比较。结果表明通过正确选择各向异性介质和添加方法,可以有效地减小三维ADI-FDTD数值色散。     

GENETIC ALGORITHM IN REDUCTION OF NUMERICAL DISPERSION OF 3-D ADI-FDTD METHOD

A new method to reduce the numerical dispersion of the three-dimensional Alternating Direction Implicit Finite-Difference Time-Domain （3-D ADI-FDTD） method is proposed. Firstly, the numerical formulations of the 3-D ADI-FDTD method are modified with the artificial anisotropy, and the new numerical dispersion relation is derived. Secondly, the relative permittivity tensor of the artificial anisotropy can be obtained by the Adaptive Genetic Algorithm （AGA）. In order to demonstrate the accuracy and efficiency of this new method, a monopole antenna is simulated as an example. And the numerical results and the computational requirements of the proposed method are compared with those of the conventional ADI-FDTD method and the measured data. In addition the reduction of the numerical dispersion is investigated as the objective function of the AGA. It is found that this new method is accurate and efficient by choosing proper objective function.     

Alternating-direction implicit (ADI) formulation of the finite-difference time-domain (FDTD) method: algorithm and material dispersion implementation

The alternating-direction implicit finite-difference time-domain (ADI-FDTD) technique is an unconditionally stable time-domain numerical scheme, allowing the /spl Delta/t time step to be increased beyond the Courant-Friedrichs-Lewy limit. Execution time of a simulation is inversely proportional to /spl Delta/t, and as such, increasing /spl Delta/t results in a decrease of execution time. The ADI-FDTD technique greatly increases the utility of the FDTD technique for electromagnetic compatibility problems. Once the basics of the ADI-FDTD technique are presented and the differences of the relative accuracy of ADI-FDTD and standard FDTD are discussed, the problems that benefit greatly from ADI-FDTD are described. A discussion is given on the true time savings of applying the ADI-FDTD technique. The feasibility of using higher order spatial and temporal techniques with ADI-FDTD is presented. The incorporation of frequency dependent material properties (material dispersion) into ADI-FDTD is also presented. The material dispersion scheme is implemented into a one-dimensional and three-dimensional problem space. The scheme is shown to be both accurate and unconditionally stable.     

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.     

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.     

等离子体的交替方向隐式时域有限差分方法

首次把交替方向隐式时域有限差分法(ADI-FDTD)推广到色散介质——无碰撞非磁化等离子体中，计算了非磁化等离子体与电磁波的相互怍用，使用ADI技术给出了无碰撞等离子体介质中的ADI-FDTD迭代公式．并解析地证明了等离子ADI-FDTD算法也是无条件稳定的，数值计算表明，等离子体ADI-FDTD算法与传统的FDTD的计算结果吻合，计算效率更高。     

A study of the numerical dispersion relation for the 2-D ADI-FDTD method

This letter presents a numerical dispersion relation for the two-dimensional (2-D) finite-difference time-domain method based on the alternating-direction implicit time-marching scheme (2-D ADI-FDTD). The proposed analytical relation for 2-D ADI-FDTD is compared with those relations in the previous works. Through numerical tests, the dispersion equation of this work was shown as correct one for 2-D ADI-FDTD.     

Improvement on the numerical dispersion of 2-D ADI-FDTD with artificial anisotropy

In this letter, by introducing artificial anisotropy into computational space, a simple and efficient approach to reduce numerical dispersion of the two-dimensional alternating direction implicit finite-difference time-domain (ADI-FDTD) method is proposed. It is shown that performance of the ADI-FDTD method can be improved significantly for both single frequency simulations and relatively wideband problems. Consequently, the usefulness and effectiveness of the ADI-FDTD method can be notably enhanced.     

Stability and Dispersion Analysis for ADI-FDTD Method in Lossy Media

The stability and dispersion analysis for the alternating-direction-implicit finite-difference time-domain (ADI- FDTD) method in lossy media is presented. Although the stability and numerical dispersion have been analyzed for the ADI-FDTD method, most of the analysis is dedicated to the cases of lossless media. Here, the stability and dispersion analysis is performed for the method in lossy media. The stability analysis theoretically proves the unconditional stability of the ADI-FDTD method in lossy media. Meanwhile, the dispersion analysis reveals the numerical loss and dispersion characteristics of this method. This will be meaningful for the evaluation and further development of the ADI-FDTD method in lossy media