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.     

求解波动方程的准粒子离散修正辛算法

针对波动方程求解,在Hamilton体系下建立了对空间离散的准粒子体系,该准粒子体系实现简单,物理意义明确;在时间离散方面,构造了一种适合高效声波模拟的修正辛格式,该格式是在常规的二阶Partitioned Runge-Kutta(PRK)基础之上构造而成,其具有三阶时间精度,从理论上分析了修正辛格式的数值稳定性和频散性能.数值结果表明,本文提出的方法在计算时间,计算精度和计算存储量等各方面性能都有相应改善。      

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.     

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.     

Canonical methods for Hamiltonian systems: numerical experiments

A Hamiltonian system possesses dynamics (e.g. preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research on this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this paper, we focus on numerical experiments that compare canonical and non-canonical numerical integrators. Test problems are taken from different areas in physical sciences. These experiments help to buttress the claims that canonical numerical integrators give results that mimic the qualitative behavior of the original system and that canonical numerical integrators are suitable for long time integrations. Our experiments indicate that higher-order canonical methods allow for larger timestep than lower-order canonical methods.     

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.     

Symplectic integrators for the Ablowitz–Ladik discrete nonlinear Schrödinger equation

The Ablowitz–Ladik discrete Nonlinear Schrödinger Eq. (IDNLS) has a noncanonical symplectic structure for which standard symplectic integrators are not applicable. Using generating functions we derive higher order symplectic schemes for the IDNLS. A comparison of the efficiency of the symplectic schemes with standard Runge–Kutta algorithms with respect to accuracy and integration time is provided.     

Multi-Symplectic Wavelet Collocation Method for Maxwell's Equations

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell's equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.     

Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations

In this paper, we propose explicit multi-symplectic schemes for Klein–Gordon–Schrödinger equation by concatenating suitable symplectic Runge–Kutta-type methods and symplectic Runge–Kutta–Nyström-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments.     

Splitting multisymplectic integrators for Maxwell’s equations

In the paper, we describe a novel kind of multisymplectic method for three-dimensional (3-D) Maxwell’s equations. Splitting the 3-D Maxwell’s equations into three local one-dimensional (LOD) equations, then applying a pair of symplectic Runge–Kutta methods to discretize each resulting LOD equation, it leads to splitting multisymplectic integrators. We say this kind of schemes to be LOD multisymplectic scheme (LOD-MS). The discrete conservation laws, convergence, dispersive relation, dissipation and stability are investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, non-dissipative, and of first order accuracy in time and second order accuracy in space. As a reduction, we also consider the application of LOD-MS to 2-D Maxwell’s equations. Numerical experiments match the theoretical results well. They illustrate that LOD-MS is not only efficient and simple in coding, but also has almost all the nature of multisymplectic integrators.     

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...     

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.     

On the stability and performance of discrete event methods for simulating continuous systems

This paper establishes a link between the stability of a first order, explicit discrete event integration scheme and the stability criteria for the explicit Euler method. The paper begins by constructing a time-varying linear system with bounded inputs that is equivalent to the first order discrete event integration scheme. The stability of the discrete event system is shown to result from the fact that it automatically adjusts its time advance to lie below the limit set by the explicit Euler stability criteria. Moreover, because it is not necessary to update all integrators at this rate, a significant performance advantage is possible. Our results confirm and explain previously reported studies where it is demonstrated that a reduced number of updates can provide a significant performance advantage compared to fixed step methods. These results also throw some light on stability requirements for discrete event simulation of spatially extended systems.     

Symplectic Runge-Kutta and related methods: recent results

Symplectic algorithms are numerical integrators for Hamiltonian systems that preserve the symplectic structure in phase space. In long time integrations these algorithms tend to perform better than their nonsymplectic counterparts. Some symplectic algorithms are derived by explicitly finding a generating function. Other symplectic algorithms are members of standard families of methods, such as Runge-Kutta methods, that just turn out to preserve the symplectic structure. Here we survey what is known about the second type of symplectic algorithms.     

Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation

The linear stability of IMEX (IMplicit–EXplicit) methods and exponential integrators for stiff systems of ODEs arising in the discrete solution of PDEs is examined for nonlinear PDEs with both linear dispersion and dissipation, and a clear method of visualization of the linear stability regions is proposed. Predictions are made based on these visualizations and are supported by a series of experiments on five PDEs including quasigeostrophic equations and stratified Boussinesq equations. The experiments, involving 24 IMEX and exponential methods of third and fourth order, confirm the predictions of the linear stability analysis, that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis. The experiments also demonstrate that IMEX methods achieve comparable stability to exponential methods, and that exponential methods are significantly more accurate only when the problem is nearly linear. Novel IMEX predictor–corrector methods are also derived.     

Symplectic and multisymplectic numerical methods for Maxwell’s equations

In this paper, we compare the behaviour of one symplectic and three multisymplectic methods for Maxwell’s equations in a simple medium. This is a system of PDEs with symplectic and multisymplectic structures. We give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments. We also show that these numerical methods preserve the divergence. Furthermore, we extend the discussion on dispersion for (multi)symplectic methods applied to PDEs with one spatial dimension, to include anisotropy when applying (multi)symplectic methods to Maxwell’s equations in two spatial dimensions. Lastly, we demonstrate how varying the Courant–Friedrichs–Lewy (CFL) number can cause the (multi)symplectic methods in our comparison to behave differently, which can be explained by the study of backward error analysis for PDEs.     

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     

Geometric formulations and variational integrators of discrete autonomous Birkhoff systems

The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.     

Improving the numerical stability of T-matrix light scattering calculations for extreme particle shapes using the nullfield method with discrete sources

Over the years the T-matrix method based on the nullfield method established itself as a fast and reliable approach for light scattering simulation. Compared to other programs based on discrete dipole approximation or finite different time domain, programs calculating the T-matrix usually are faster. Unlike the Mie theory the underlying nullfield method can also be used for non-spherical particles, but one will observe a limited numerical stability in this case. Using an advanced approach, the nullfield method with discrete sources, we would like to demonstrate how to improve the numerical stability and to get results also for particles with extreme shapes like fibers or discs with high aspect ratios or concavities. In this paper we intentionally keep the mathematical part rather small, instead we focus on more general explanations for users of corresponding computer programs by outlining basic ideas and concepts.     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.