含时Schrödinger方程的高阶辛FDTD算法研究

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离散辛框架;探讨了高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算例表明:当空间采用高阶同位差分格式时, 辛积分可提高算法的稳定度;SFDTD(3, 4)法和FDTD(2, 4)法较传统的FDTD(2, 2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明: SFDTD(3, 4)法较传统的FDTD(2, 2)法及高阶FDTD(2, 4)法有着更好的计算精度和收敛性, 且SFDTD(3, 4)法能够保持量子系统的能量守恒, 适用于长时间仿真.     

含时Schrdinger方程的高阶辛FDTD算法研究

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离散辛框架;探讨了高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算例表明:当空间采用高阶同位差分格式时,辛积分可提高算法的稳定度;SFDTD(3,4)法和FDTD(2,4)法较传统的FDTD(2,2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明:SFDTD(3,4)法较传统的FDTD(2,2)法及高阶FDTD(2,4)法有着更好的计算精度和收敛性,且SFDTD(3,4)法能够保持量子系统的能量守恒,适用于长时间仿真.     

力梯度辛方法在圆型限制性三体问题中的应用

旋转坐标系下的圆型限制性三体问 题因含非惯性系所附加的影响部分使得动能不是动量的严格二次型, 可能导致力梯度辛积分算法的应用遇到困难. 从Lie算子运算出发, 严格论证了力梯度算子在这种情形下的物理意义 仍然像质心惯性坐标系下的圆型限制性三体问题那样是引力的梯度, 而不是引力与非惯性力所得合力的梯度, 表明了力梯度辛方法适合求解旋转坐标系下的圆型限制性三体问题. 通过应用四阶力梯度辛方法、最优化四阶力梯度辛方法和Forest-Ruth 辛方法分别求解该问题, 进行了数值对比研究, 结果显示最优化型力梯度算法能够取得最好精度. 还应用最优化型算法计算两邻近轨道的Lyapunov指数和快速Lyapunov指标, 确保高精度辛方法能够贯穿于这些混沌指标计算的全过程, 以便准确刻画此系统的动力学定性性质. 关键词： 辛积分器 圆型限制性三体问题 混沌 Lyapunov指数     

含时Schroedinger方程的高阶辛FDTD算法研究

提出了一种新的算法一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离散辛框架;探讨了高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算例表明：当空间采用高阶同位差分格式时,辛积分可提高算法的稳定度;SFDTD（3,4）法和FDTD（2,4）法较传统的FDTD（2,2）法数值色散性明显改善．对二维量子阱和谐振子的仿真结果表明：SFDTD（3,4）法较传统的FDTD（2,2）法及高阶FDTD（2,4）法有着更好的计算精度和收敛性,且SFDTD（3,4）法能够保持量子系统的能量守恒,适用于长时间仿真．     

逼近积分点数下限的五阶容积卡尔曼滤波定轨算法

为了在保持滤波定轨精度不变的条件下提高定轨计算的实时性,提出一种新的逼近积分点个数下限的五阶容积卡尔曼滤波定轨算法.首先,采用一种数值容积准则对非线性函数的高斯加权积分进行近似,该准则所需的积分点个数仅比五阶代数精度容积准则积分点个数的理论下限多一个积分点,并在贝叶斯滤波算法框架下推导出本文算法的更新步骤.然后,给出实时定轨所需的状态方程和量测方程,在状态方程中考虑了J2项引力摄动和大气阻力摄动,在量测方程中利用坐标系转换推导了轨道状态与测量元素之间的非线性关系.仿真实验结果表明,本文所提算法在定轨精度方面与已有的五阶滤波算法相当,但所需的积分点个数最少,计算实时性最高,从而验证了本文算法的有效性.     

算法对混沌体系逃逸率的影响

以Hénon-Heiles体系为例，研究算法对混沌体系运动轨道和逃逸率计算结果的影响.比较了新发现的四阶辛算法和一种非辛的高阶算法得到的结果.发现两种算法给出的轨迹之间的距离随时间增大，增加的速度可以作为体系相空间混沌的度量.通过跟踪大数量的粒子轨迹，提取出了逃逸率随体系能量的变化.发现由两种算法得到的逃逸率相互符合得很好. 关键词： 逃逸率 Hénon-Heiles体系 辛算法     

高阶辛算法在典型量子器件模拟中的应用

利用辛积分和高阶交错差分方法建立了求解含时薛定谔方程的高阶辛算法(SFDTD(4,4)).对空间部分的二阶导数采用四阶准确度的差分格式离散得到随时间演化的多维系统再引入四阶辛积分格式离散;探讨了SFDTD(4,4)法的稳定性,获得了含时薛定谔方程的一维以及多维的稳定性条件,并得到在含势能情况下该稳定性条件的具体表达式;借助复坐标沿伸概念,实现了SFDTD(4,4)法在量子器件模拟中的完全匹配层吸收边界条件.结合一维量子阱和金属场效应管传输的仿真,结果表明较传统的时域有限差分算法,SFDTD(4,4)有着更好的计算准确度,适用于长时间仿真.算法及相关结果可为实际量子器件的设计提供必要的参考.     

一维强场模型研究中的非齐线性正则方程的辛算法

就一维强场模型,采用对称差商代替空间变量的2阶偏导数,将含有SchrÖdinger方程的初边值问题离散成"非齐线性正则方程",它的齐方程的通解和非齐方程特解都由"辛变换生成",分别采用辛格式计算.采用这种辛算法和R-K法计算了一个数值例子,并与精确解作了比较.结果表明,经长时间计算后,辛算法保持解的固有特征,而R-K法则面目全非.     

任意阶显式精细积分多步法的常用形式及其高阶次数值计算

基于任意阶显式精细积分多步法的一般公式,给出其几种常用形式,并实现了高阶次数值计算,将新算法应用于射线方程和双原子系统经典轨迹数值计算中.数值计算结果表明任意阶显式精细积分多步法是一种高精度、高效率、稳定性较好的方法,并且可方便地进行高阶次的运算.     

高阶辛算法的稳定性与数值色散性分析

利用Maxwell方程的哈密尔顿函数,导出对应的欧拉-哈密尔顿方程.利用辛积分技术与高阶交错差分技术,建立求解三维时域Maxwell方程的高阶辛算法;结合电磁场中的物理概念,借助矩阵分析和张量分析理论,获得高阶时域方法及高阶辛算法的稳定性和数值色散性的统一处理新方法.用数值结果证实方法的正确性,与FDTD算法和其它时域高阶方法相比,高阶辛算法具有较大的计算优势,为电磁计算提供了新的途径.     

Energy Conserving, Liouville, and Symplectic Integrators

In this paper, we construct an integrator that converves volume in phase space. We compare the results obtained using this method and a symplectic integrator. The results of our experiments do not reveal any superiority of the symplectic over strictly volume-preserving integrators. We also investigate the effect of numerically conserving energy in a numerical process by rescaling velocities to keep energy constant at every step. Our results for Henon-Heiles problem show that keeping energy constant in this way destroys ergodicity and forces the solution onto a periodic orbit.     

Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields

A variational symplectic integrator for the guiding-center motion of charged particles in general magnetic fields is developed for long-time simulation studies of magnetized plasmas. Instead of discretizing the differential equations of the guiding-center motion, the action of the guiding-center motion is discretized and minimized to obtain the iteration rules for advancing the dynamics. The variational symplectic integrator conserves exactly a discrete Lagrangian symplectic structure, and has better numerical properties over long integration time, compared with standard integrators, such as the standard and variable time-step fourth order Runge-Kutta methods.     

Klein-Gordon-Schr(o)dinger方程的辛Fourier拟谱格式

主要讨论Klein-Gordon-Sehrodinger方程的Fourier拟谱辛格式,包括中点公式和Stormer/Vedet格式.首先构造一个哈密尔顿方程,针对此哈密尔顿方程,在空间方向用Fourier拟谱离散得到一个有限维的哈密尔顿系统,对此有限维系统在时间方向用St(o)rmer/Verlet方法离散得到KGS...     

半隐Euler法和隐中点法嵌入混合辛积分器的比较

当Hamilton函数分解为可积和不可积两部分时,前者能用分析方法给出解析解,而后者可借助一阶半隐Euler法或二阶隐中点法等数值求解,将这种解析和数值解法组合能构造二阶混合辛积分器.理论分析表明Euler嵌入法的稳定区要小于中点嵌入法的.再分别以圆形限制性三体问题和相对论自旋致密双星后牛顿Hamilton构型为例,详细比较了两嵌入法的性能特点.二者的数值精度、稳定性及计算效率与Hamilton的分解方式和轨道类型有关.就圆形限制性三体问题而言,当Hamilton采用势能和含坐标与动量混合项在内的动能分解 关键词： 辛积分器 后牛顿近似 自旋致密双星 混沌     

Symplectic integrators from composite operator factorizations

I derive fourth order symplectic integrators by factorizing the exponential of two operators in terms of an additional higher order composite operator with positive coefficients. One algorithm requires only one evaluation of the force and one evaluation of the force and its gradient. When applied to Kepler's problem, the energy error function associated with these algorithms are approximately 10 to 80 times smaller than the fourth order Forest-Ruth, Candy-Rozmus integrator.     

A discrete action principle for electrodynamics and the construction of explicit symplectic integrators for linear,non-dispersive media

In this work, we derive a discrete action principle for electrodynamics that can be used to construct explicit symplectic integrators for Maxwell’s equations. Different integrators are constructed depending on the choice of discrete Lagrangian used to approximate the action. By combining discrete Lagrangians in an explicit symplectic partitioned Runge–Kutta method, an integrator capable of achieving any order of accuracy is obtained. Using the von Neumann stability analysis, we show that the integrators greatly increase the numerical stability and reduce the numerical dispersion compared to other methods. For practical purposes, we demonstrate how to implement the integrators using many features of the finite-difference time-domain method. However, our approach is also applicable to other spatial discretizations, such as those used in finite element methods. Using this implementation, numerical examples are presented that demonstrate the ability of the integrators to efficiently reduce and maintain a minimal amount of numerical dispersion, particularly when the time-step is less than the stability limit. The integrators are therefore advantageous for modeling large, inhomogeneous computational domains.     

Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.     

Multisymplectic Pseudospectral Discretizations for (3+1)-Dimensional Klein-Gordon Equation

We explore the multisymplectic Fourier pseudospectral discretizations for the (3+1)-dimensional Klein-Gordon equation in this paper. The corresponding multisymplectic conservation laws are derived. Two kinds of explicit symplectic integrators in time are also presented.