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TN82007021058辛算法的稳定性及数值色散性分析/黄志祥,吴先良(安徽大学计算智能与信号处理教育部重点实验室)//电子学报.―2006,34(3).―535～538.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     

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.     

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.     

基于高阶辛算法的纳米器件本征问题仿真

研究精确和高效的数值方法是现代纳米器件建模和优化的重要目标之一,而分析大部分纳米器件特性的切入点是确定器件结构的能量本征值和能量本征态。本文提出了一种新的算法—高阶辛时域有限差分法(SFDTD(3,4): symplectic finite-difference time-domain)求解含时薛定谔方程。在时间上采用三阶辛积分格式离散,空间上采用四阶精度的同位差分格式离散,建立了求解含时薛定谔方程的高阶辛时域有限差分算法。将高阶辛算法SFDTD(3,4)用于一维量子阱中盒中粒子和谐振子的仿真中,实验结果表明SFDTD(3,4)法比传统的时域有限差分算法以及高阶时域有限差分算法更加准确,适用于对纳米器件本征问题的长时间仿真。     

Maxwell 方程组的多辛算法

Maxwell方程在线性、各向同性、均匀、无源的介质中具有自然的多辛结构,可以表示为多辛Hamilton系统。Maxwell方程的多辛算法即对Maxwell方程在时间、空间同时进行保辛离散得到相应的差分格式。文中给出了5种麦克斯韦方程的多辛算法,分析并比较了这5种方法的数值色散特性。数值计算结果表明这些算法能很好地保持Maxwell方程的离散全局能量守恒特性。     

基于辛算法的二维电磁场散射问题的研究

辛算法是保持Hamilton系统辛结构的一种新的数值方法,由于 Maxwell方程是一无穷维Hamilton系统,因此可将辛算法用于电磁场模拟中.本文提出一种基于辛分块Runge-Kutta(PRK)方法的显式辛算法,并将它成功应用于二维电磁散射问题的计算中.通过对金属方柱散射场的数值模拟,比较了FDTD法和低阶辛算法(一阶和二阶),结果表明低阶辛算法不仅与FDTD法精度相当,而且可以减少存储空间和计算时间,尤其是一阶辛算法节省了大约的CPU时间,提高了计算速度,体现了该算法的优越性.     

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ADI-FDTD算法的数值色散分析

该文给出高阶交替方向隐时域优先差分(ADI-FDTD)算法,即在ADI-FDTD迭代公式的基础上对时间的差分仍然采用二阶中心差分格式,而对空间的差分则采用四阶中心差分格式,并解析地证明了所给出的高阶ADI-FDTD算法仍然满足无条件稳定方程,同时对增长因子相位的分析,得到数值色散关系,最后对其数值色散误差进行了分析,研究表明与普通ADI-FDTD相比,其色散误差较小。     

碰撞等离子体的高阶FDTD算法

给出了电磁波在均匀、碰撞等离子体中传播的四阶时间和四阶空间FDTD算法.该算法比Yee氏FDTD算法每一个网格每一维增加一个存储单元,与常规的二阶等离子体FDTD算法相同.由于采用四阶时间和四阶空间近似,因此该算法能有效地减小数字色散误差,其频带宽度比二阶算法的频带宽度更宽.为了验证该高阶算法的正确性,对均匀、碰撞等离子体平板的电磁波反射系数进行了计算,并与解析结果、二阶FDTD计算结果进行了比较,证明了该算法的高效和精确.     

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。     

Higher Stability Limits for the Symplectic FDTD Method by Making Use of Chebyshev Polynomials

A new scheme is introduced for obtaining higher stability performance for the symplectic finite-difference time-domain (FDTD) method. Both the stability limit and the numerical dispersion of the symplectic FDTD are determined by a function zeta. It is shown that when the zeta function is a Chebyshev polynomial the stability limit is linearly proportional to the number of the exponential operators. Thus, the stability limit can be increased as much as possible at the cost of increased number of operators. For example, the stability limit of the four-exponential operator scheme is 0.989 and of the eight-exponential operator scheme it is 1.979 for fourth-order space discretization in three dimensions, which is almost three times the stability limit of previously published symplectic FDTD schemes with a similar number of operators. This study also shows that the numerical dispersion errors for this new scheme are less than those of the previously reported symplectic FDTD schemes     

基于传播子技术的辛时域多分辨率方法

 数值求解三维时域Maxwell方程的过程中,保持方程的内在结构显得尤为重要.利用Hamilton函数的变分形式,将Maxwell方程表述为Hamilton正则方程形式.在时域方向,利用辛传播子技术对方程进行离散以保持方程的内在结构;在空域方向,采用时域多分辨率方法对三维旋度算符进行差分离散,建立了求解Maxwell方程的辛时域多分辨率(S-MRTD)方法.对S-MRTD方法的稳定性及数值色散性进行了系统的探讨,数值结果表明该方法的正确性及高精度性.     

一种高阶辛时域有限差分法的研究

从电磁场方程的Hamilton函数出发,提出了一种基于辛时域积分技术的高阶时域有限差分方法.该方法对时域的离散采用了能够保证系统的相空间体积不变和总能量不变的辛格式,对于空间的离散采用中心差分格式.计算结果表明与传统的时域高阶差分方法--Runge-Kutta法比较,该方法计算速度和计算精度都有较大的提高.     

Survey on Symplectic Finite-Difference Time-Domain Schemes for Maxwell's Equations

To discretize Maxwell's equations, a variety of high-order symplectic finite-difference time-domain schemes, which use th-order symplectic integration time stepping and th-order staggered space differencing, are surveyed. First, the order conditions for the symplectic integrators are derived. Second, the comparisons of numerical stability, dispersion, and energy-conservation are provided between the high-order symplectic schemes and other high-order time approaches. Finally, these symplectic schemes are studied by using different space and time strategies. According to our survey, high-order time schemes for matching high-order space schemes are required for optimum electromagnetic simulation. Numerical experiments have been conducted on radiation of electric dipole and wideband S-parameter extraction of dielectric-filled waveguide. The results demonstrate that the high-order symplectic scheme can obtain satisfying numerical solutions under high Courant-Friedrichs-Levy number and coarse grid conditions.     

Higher-order finite-difference schemes for electromagneticradiation, scattering, and penetration .1. Theory

Higher-order schemes for the finite-difference time-domain (FDTD) method are presented, in particular, a second-order-in-time, fourth-order-in-space method: FDTD(2,4). This method is compared to the original Yee (1966) FDTD scheme. One-dimensional update equations are presented, and the characteristics of the FDTD(2,4) scheme are investigated. Theoretical results for numerical stability and dispersion are presented, with numerical results for the latter, as well. The use of the perfectly matched layer for the FDTD(2,4) scheme is discussed, and numerical results are shown     

A generalized higher order finite-difference time-domain method and its application in guided-wave problems

In this paper, a (2M,4) scheme of the finite-difference time-domain (FDTD) method is proposed, in which the time differential is of the fourth order and the spatial differential using the discrete singular convolution is of order 2M. Compared with the standard FDTD and the scheme of (4, 4), the scheme of (2M, 4) has much higher accuracy. By choosing a suitable M/spl ges/2, the (2M, 4) scheme can arrive at the highest accuracy. In addition, an improved approximation of the symplectic integrator propagator is presented for the time differential. On the one hand, it can directly simulate unlimited conducting structures without the air layer between the perfectly matched layer and inner structure; on the other hand, it needs only a quarter of the memory space required by the Runge-Kutta time scheme and requires one third of the meshes in every direction of the standard FDTD method. By choosing suitable meshes and bandwidth M, our scheme not only retains higher accuracy but also saves memory space and CPU time. Numerical examples are provided to show the high accuracy and effectiveness of the proposed scheme.     

辛时域多分辨率算法在波导结构仿真中的应用

文中基于Daubechies紧支撑尺度函数的辛时域多分辨率(symplectic multiresolution time-domain, S-MRTD)算法,在时间方向上采用优化的3级3阶的辛算子进行离散,以减少时间步的迭代次数,在空间方向上采用小波尺度函数展开算法进行离散. 给出了S-MRTD算法的三维迭代公式,并将其应用到波导结构相关特征参数的数值计算和分析中. 数值计算结果表明,与传统的MRTD算法相比,S-MRTD算法在使用粗网格和较高的稳定性常数时,仍能得到精确计算结果,且具有内存使用少、计算效率高的特点.     

Efficient NL-FDTD Solution Schemes for the Phase-Sensitive Second Harmonic Generation Problem

A combination of a higher order accurate FDTD algorithm, a decoupling procedure, and a moving computational window is presented for the solution of the phase-sensitive second harmonic generation problem. The requirement that the spatial step size in the propagation direction be a small fraction of the wavelength is significantly relaxed using the proposed efficient FDTD schemes. It has been shown that these fully explicit schemes deliver convergence of the solution using significantly less computation time and less memory requirement as compared to the standard FDTD scheme.     

A higher order (2,4) scheme for reducing dispersion in FDTDalgorithm

A finite-difference time-domain (FDTD) scheme with second-order accuracy in time and fourth-order in space is discussed for the solution of Maxwell's equations in the time domain. Compared with the standard Yee (1966) FDTD algorithm, the higher order scheme reduces the numerical dispersion and anisotropy and has improved stability. Dispersion analysis indicates that the frequency band in which the higher order scheme yields an accurate solution is widened on the same grid, this means a larger space increment can be chosen for the same excitation. Numerical results show the applications of the scheme in modeling wide-band electromagnetic phenomena on a coarse grid