一种优化参数的LOD-FDTD算法及其数值特性

提出一种改进的参数优化局部一维时域有限差分(LOD-FDTD)方法,该方法将时间步长等分成3步,沿坐标方向加上色散控制因子,以降低数值色散误差。本文首先证明改进方法的稳定性,并分析其数值色散误差。结果表明改进方法的数值色散误差小于传统的LOD-FDTD方法。     

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.     

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。     

三维LOD-FDTD方法在PMC边界处的精确格式

证明了局部一维时域有限差分(LOD-FDTD)方法实现理想磁导体 (PMC)边界时的待求场分量系数与传统的LOD-FDTD方法系数不同。通过在获得该系数前应用理想导体边界条件,得到对应的修正系数。计算了单个PMC立方体和对称的两个PMC立方体的双站RCS。计算结果表明,PMC边界作为理想导体表面时,传统LOD-FDTD方法计算误差较大,采用修正系数的计算结果与传统FDTD方法计算结果更为吻合;PMC边界作为截断计算空间的对称面,采用修正系数的计算结果与传统LOD-FDTD方法计算结果相同。采用修正系数处理PMC边界无需区分PMC边界是理想磁导体表面还是截断计算空间的对称面,具有统一的表达式,计算理想磁导体表面较传统LOD-FDTD方法误差更小。     

高阶FDTD法分析电-大尺寸光波导器件

高阶时域有限差分(FDTD)法用于电-大尺寸平面光波导器件的时域分析,实现了高阶FDTD法的理想匹配层(PML)吸收边界条件;研究了高阶FDTD法的数值色散特性,并对平行介质带定向耦合器进行了数值模拟,所得结果与解析解非常一致。     

一种新的数学方法及其在色散缓变光纤方程中的应用

将一种新而简洁的数学方法应用于色散缓变光纤中含高阶色散的非线形薛定谔方程(NLSE)中,分析了负三阶色散和负四阶色散对孤子压缩效应和频移效应的影响,并给出了压缩长度的数学表达式.这些表达式所预示的规律,或与已知的实验结果一致,或与数值模拟的结果一致.     

光子晶体光纤超连续谱高阶孤子分裂的数值模拟

基于高阶非线性薛定谔方程,利用分布傅里叶方法（SSFM）数值模拟了高阶孤子在光子晶体光纤反常色散区中的传输。从脉冲形状和频谱两个方面分析讨论了三阶色散、自陡峭效应和脉冲内拉曼散射对高阶孤子传输的影响。计算结果表明：高阶效应使高阶孤子的演化不再具有周期性特性。其中,对高阶孤子分裂起主导作用的是三阶色散和孤子内拉曼散射。高阶效应均使最终频谱得到了不同程度的展宽。     

基于厄米-高斯函数法研究1维光子晶体波导

为了对折射率型1维光子晶体缺陷波导中的光传输进行有效数值模拟,采用此类型波导的厄米-高斯函数展开方法进行了研究。首先给出了计算方法的详细理论推导,然后利用该方法计算了偏振态不同、结构参量不同的情况下波导本征模式的色散关系、模场空间分布、能量控制因子及等效折射率。结果表明,1维光子晶体缺陷波导与阶跃平面光波导主要差别在于高阶模式,且可通过调节1维光子晶体的结构参量来有效调控高阶模式的传输。     

高阶有限元-边界积分法在二维散射中的应用

以二阶、三阶基函数为例,应用高阶有限元-边界积分法分析了二维散射体电磁散射特性。计算了几种二维方柱(导体和介质)的雷达散射截面,结果与矩量法一致,对三种数值结果进行了误差分析。结果表明:高阶有限元-边界积分法比一阶有限元-边界积分法有着更高的计算精度、收敛速度和计算效率。     

一种提高内存使用效率的时域有限差分算法

证明了即使在无源区域,局部一维时域有限差分法(LOD-FDTD)所给出的电磁场量也不满足零散度关系,推导了该散度关系的具体表达式。基于该非零散度关系和麦克斯韦旋度方程,将LOD-FDTD法与减缩时域有限差分法(R-FDTD)相结合,得到一种新的局部一维减缩时域有限差分法(LOD-R-FDTD)。该方法不仅具有LOD-FDTD方法的优势,计算公式简单,消除了CFL稳定条件对时间步长的限制,而且与LOD-FDTD相比平均节约了1/3内存使用量。通过仿真计算与其他方法对比,证明了LOD-R-FDTD方法的准确性和有效性。     

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.      

Numerical Dispersion Analysis With an Improved LOD–FDTD Method

In this letter, a modified locally one-dimensional finite-difference time-domain (LOD-FDTD) method is proposed. The dispersion behavior is investigated and compared with the conventional LOD-FDTD method. It is found that for a Courant-Friedrich-Levy number equal to 5 the modified LOD-FDTD method performs approximately 20% better than the conventional LOD-FDTD method     

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.      

A path integral time-domain method for electromagnetic scattering

A new full wave time-domain formulation for the electromagnetic field is obtained by means of a path integral. The path integral propagator is derived via a state variable approach starting with Maxwell's differential equations in tensor form. A numerical method for evaluating the path integral is presented and numerical dispersion and stability conditions are derived and numerical error is discussed. An absorbing boundary condition is demonstrated for the one-dimensional (1-D) case. It is shown that this time domain method is characterized by the unconditional stability of the path integral equations and by its ability to propagate an electromagnetic wave at the Nyquist limit, two numerical points per wavelength. As a consequence the calculated fields are not subject to numerical dispersion. Other advantages in comparison to presently popular time-domain techniques are that it avoids time interval interleaving and it does not require the methods of linear algebra such as basis function selection or matrix methods     

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.      

ADI-MRTD算法的数值色散性分析

针对传统的时域多分辨分析(MRTD)方法的稳定性不足问题,讨论了一种将交替方向隐式技术(ADI)与MRTD算法相结合的交替方向隐式时域多分辨分析算法(ADI-MRTD)。导出了基于Daubechies小波尺度函数的ADI-MRTD算法的差分公式和色散性方程,同时证明了其仍然满足无条件稳定方程。并讨论了空间步长、时间步长和电磁波传播方向等因素对ADI-MRTD算法的数值色散影响。结果表明:ADI-MRTD算法的数值色散特性优于传统的时域有限差分(FDTD)算法。     

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     

Unconditionally Stable LOD–FDTD Method for 3-D Maxwell's Equations

This letter presents an unconditionally stable locally 1-D finite-difference time-domain (LOD-FDTD) method for 3-D Maxwell's equations. The method does not exhibit the second-order noncommutativity error and its second-order temporal accuracy is ascertained via numerical justification. The method also involves simpler updating procedures and facilitates exploitation of parallel and/or reduced output processing. This leads to its higher computation efficiency than the alternating direction implicit and split-step FDTD methods