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.     

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.     

辛几何算法在计算非磁化等离子体中波传播轨迹时的应用

为了在数值计算中保持哈密顿系统的辛几何结构不变,利用辛几何算法得到了在线性哈密顿系统中射线追踪方程的一般辛差分格式。通过具体算例,利用辛几何算法计算了波在非磁化等离子体中的传播轨迹,并且与传统Runge-Kutta-Fehlberg算法所得结果进行了比较。利用辛几何算法所得传播轨迹与解析解一致,其色散函数值的误差随时间线性增长,能在长时间内保持色散函数值在一个很小的误差范围内。利用传统的Runge-Kutta-Fehlberg算法所得传播轨迹与解析解不一致,其误差随时间做大幅振荡增加。计算结果表明辛几何算法在保持传播轨迹和色散函数值方面具有独特的优势。     

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.     

Dynamics of Charged Particles Around a Magnetized Black Hole with Pseudo–Newtonian Potential

The linear stability of equilibria of charged particles moving near a compact object with a dipole magnetic field and a pseudo-Newtonian potential is analyzed detailedly.An optimal fourth-order force gradient symplectic method,as a global symplectic integrator that can simultaneously solve both the equations of motion and the variational equations,is used to calculate fast Lyapunov indicators.In this way,dynamical structures are described,and parameter domains for causing chaos are found.     

Integrable systems and invariant curve flows in centro-equiaffine symplectic geometry

In this paper, the theory for curves in centro-equiaffine symplectic geometry is established. Integrable systems satisfied by the curvatures of curves under inextensible motions in centro-equiaffine symplectic geometry are identified. It is shown that certain non-stretching invariant curve flows in centro-equiaffine symplectic geometry are closely related to the matrix KdV equations and their extension.     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

Discretizing families of linearizable equations

We investigate generalizations of the discrete Riccati equation as a linearizable system, to multicomponent linearizable systems. These are discretizations of nonlinear ordinary differential equations with superposition formulas. We present discrete matrix Riccati equations, projective, conformal, orthogonal and symplectic Riccati equations. Also obtained are discrete equations, based on complex orthogonal and symplectic groups, that in the continuous limit involve fourth order polynomial nonlinearities. All these equations satisfy the criterion of singularity confinement.     

两个三阶最优化力梯度辛积分器的对称组合

利用已存在的三阶最优化力梯度辛格式以对称组合方法获得两个新的四阶力梯度辛积分器.它们在求解摄动Kepler混沌问题的能量精度和一维定态Schrö,dinger方程的能量本征值精度方面比Forest-Ruth四阶非力梯度辛积分器要好得多,甚至还要明显优越于已有的四阶最优化力梯度辛积分器.     

Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.     

An improved symplectic precise integration method for analysis of the rotating rigid–flexible coupled system

This paper presents an improved symplectic precise integration method (PIM) to increase the accuracy and keep the stability of the computation of the rotating rigid–flexible coupled system. Firstly, the generalized Hamilton's principle is used to establish a coupled model for the rotating system, which is discretized and transferred into Hamiltonian systems subsequently. Secondly, a suitable symplectic geometric algorithm is proposed to keep the computational stability of the rotating rigid–flexible coupled system. Thirdly, the idea of PIM is introduced into the symplectic geometric algorithm to establish a symplectic PIM, which combines the advantages of the accuracy of the PIM and the stability of the symplectic geometric algorithm. In some sense, the results obtained by this method are analytical solutions in computer for a long span of time, so the time-step can be enlarged to speed up the computation. Finally, three numerical examples show the stability of computation, the accuracy of solving stiff equations and the capability of solving nonlinear equations, respectively. All these examples prove the symplectic PIM is a promising method for the rotating rigid–flexible coupled systems.     

Analysis of numerical dispersion properties of SFDTD and MRTD schemes

In this paper, the Maxwell's equations are written as Hamilton canonical equations by using Hamilton functional variation method. Maxwell's equations can be discretized with symplectic propagation technique combined with high-order difference schemes approximations to construct symplectic finite difference time domain (SFDTD) method. The high-order dispersion equations of the scheme for space is deduced. The numerical dispersion analysis is included, and it is compared with the multiresolution time-domain (MRTD) method based on the Daubechies scaling functions. Numerical results show high efficiency and accuracy of the SFDTD method.     

Symplectic structures,their Bäcklund transformations and hereditary symmetries

It is shown that compatible symplectic structures lead in a natural way to hereditary symmetries. (We recall that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetries all generated by this hereditary symmetry. Furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations). This result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations.In addition we show how symplectic structures transform under Bäcklund transformations. This leads to a method for generating a whole class of symplectic structures from a given one.Several examples and applications are given illustrating the above results. Also the connection of our results with those of Gelfand and Dikii, and of Magri is briefly pointed out.     

The Hamiltonian structure of the Maxwell-Vlasov equations

Morrison [25] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison's does not [37]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid dynamics.     

孤立波方程的保结构算法

讨论了孤立波方程的保结构差分算法，以一些经典的孤立波方程为例，如KdV，sine-Gordon，K-P方程，给出了它们的辛和多辛结构，说明辛和多辛算法的可适用性．提出局部守恒算法和广义保结构算法的概念，它们是保结构算法的概念自然推广．还给出一种能系统构造局部守恒格式的复合方法．数值例子说明，保结构数值能很好模拟各种孤立波的演化。     

Structure-preserving algorithms for Birkhoffian systems

We propose a new approach to construct structure-preserving algorithms for Birkhoffian systems. First, the Pfaff–Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Then, taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhoffian systems. Simulation results of the given example indicate that structure-preserving algorithms obtained by this method have great advantage in conserving conserved quantities.     

电磁波导的辛分析与对偶棱边元

将电磁波导的控制方程导向了Hamilton体系、辛几何的形式.以电磁场的横向分量组成对偶向量并采用分离变量法，可以得到Hamilton算子矩阵的辛本征值问题.共轭辛正交归一关系、辛本征解展开定理等均可在此应用.对于复杂横截面和填充非均匀材料的电磁波导，提出对偶棱边元，对截面半解析离散后即可进行数值求解.对偶棱边元克服了结点基有限元求解电磁场问题的困难，与常规棱边元相比在某些方面具有一定的优势. 关键词： 电磁波导 Hamilton体系 对偶变量 棱边元     

Birkhoffian Symplectic Scheme for a Quantum System

In this paper, a classical system of ordinary differential equations is built to describe a kind of n-dimensional quantum systems. The absorption spectrum and the density of the states for the system are defined from the points of quantum view and classical view. From the Birkhoffian form of the equations, a Birkhoffian symplectic scheme is derived for solving n-dimensional equations by using the generating function method. Besides the Birkhoffian structure- preserving, the new scheme is proven to preserve the discrete local energy conservation law of the system with zero vector f . Some numerical experiments for a 3-dimensional example show that the new scheme can simulate the general Birkhoffian system better than the implicit midpoint scheme, which is well known to be symplectic scheme for Hamiltonian system.     

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.