保结构算法的相位误差分析及其修正

辛算法和保能量算法是应用最为广泛的两种保结构算法.本文从相位误差的角度给出了他们的比较结果.我们针对线性动力系统,分别分析了基于Pade对角逼近给出的辛算法和基于平均向量场法得到的能量守恒算法的相位误差,并通过数值验证了分析结果.文章还给出了保结构算法相位误差的改进方法,并通过数值例子验证了方法的有效性.     

求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析

对求解大规模稀疏Hamilton矩阵特征问题的辛Lanczos算法给出了舍入误差分析.分析表明辛Lanczos算法在无中断时,保Hamilton结构的限制没有破坏非对称Lanczos算法的本质特性.本文还讨论了辛Lanczos算法计算出的辛Lanczos向量的J一正交性的损失与Ritz值收敛的关系.结论正如所料,当某些Ritz值开始收敛时.计算出的辛Lanczos向量的J-正交性损失是必然的.以上结果对辛Lanczos算法的改进具有理论指导意义.     

基于径向基逼近的KdV方程的无网格辛算法

基于径向基逼近理论,本文为KdV方程构造了一个无网格辛算法.首先借助径向基空间离散Hamilton函数以及Poisson括号,把KdV方程转化成一个有限维的Hamilton系统.然后用辛积分子离散有限维系统,得到辛算法.文章进一步讨论了所构造辛算法的收敛性和误差界.数值例子验证了理论分析.     

精细辛几何算法的误差估计

该文讨论了精细辛几何算法的计算误差,先展开二阶和四阶精细辛几何算法的表达式得到误差同精细剖分数目的关系,然后分析了任意阶精细辛几何算法的误差,得到了一致简洁的结果,总的误差可近似表示为单个精细步长的误差乘以剖分数目,最后讨论了在要求控制精度下剖分数目的选取,该方法克服了算法精度对积分时间步长的依赖性.     

基于Hamilton体系和辛算法的微分对策数值法

微分对策求解往往涉及到困难的两点边值问题（TPBV）,将线性二次型微分对策问题归结于Hamilton体系．对Hamilton系统,辛几何算法具有能复制Hamilton系统的动态结构并保持相平面上的测度的优点．从Hamilton系统角度,探讨了线性二次型微分对策系统的辛性质;作为尝试,对无限期间线性二次型微分对策的计算引入Symplectic-Runge-Kutta算法．给出了一个数值计算实例,从结果可以说明这种方法的可行,也体现了辛算法对系统的能量具有良好的守恒性．     

自治Birkhoff系统的广义正则变换和辛算法研究

研究了自治Birkhoff系统的广义正则变换,将Hamilton系统的辛算法推广到Birkhoff系统,通过引入凯莱变换和生成函数法构造Birkhoff方程的Birkhoff的辛差分格式,同时讨论了Birkhoff差分格式的辛算法.     

太阳帆塔轨道和姿态耦合动力学建模及辛求解

在建立太阳帆塔太阳能电站简化模型的基础上,将系统的动力学方程从Lagrange体系导入到了Hamilton体系,给出了带约束的Hamilton正则方程;进而采用祖冲之类算法和辛Runge-Kutta方法分析了太阳帆塔轨道和姿态耦合系统的动力学特性,并讨论了算法的保能量、保约束特性;最后,数值模拟了系统的动力学特性,说明了所提方法的有效性.     

Sine-Gordon方程的高阶保能量算法

正1引言具有能量守恒、辛结构等固有特性的Hamilton系统被广泛地用于描述各种物理现象,并在自然界中具有普遍性.构造保持Hamil‘ton系统的固有特性的数值算法,对于正确求解Hamilton系统具有重要的意义.冯康院士及其研究小组提出了保持Hamilton系统辛结构的辛几何算法~([1-3]),辛几何算法凭借其优异的稳定性和长时间计算能力.被广泛应用于孤立子方程,流体力学和量子系统等的计算中~([4-5]).然而,向后误差分析表明~([6-7]),辛算法只能近似保持:Hamilton系统能量守恒特性.     

Hamilton系统的连续有限元法

利用常微分方程的连续有限元法,对非线性Hamilton系统证明了连续一次、二次有限元法分别是2阶和3阶的拟辛格式,且保持能量守恒;连续有限元法是辛算法对线性Hamilton系统,且保持能量守恒．在数值计算上探讨了辛性质和能量守恒性,与已有的辛算法进行对比,结果与理论相吻合．     

保辛-保能的数值积分

针对Hamilton动力系统时变非线性问题,应用混合能变分原理,提出Hamilton系统的离散积分保辛算法.在此基础上,对Hamilton系统引入参变量,设计非线性问题迭代算法格式,通过对参变量的调整,在积分格点上实现了Hamilton系统数值积分保辛同时保能的目标.     

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ω h (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.     

The rate of error growth in Hamiltonian-conserving integrators

In this note, we consider numerical methods for a class of Hamiltonian systems that preserve the Hamiltonian. We show that the rate of growth of error is at most linear in time when such methods are applied to problems with period uniquely determined by the value of the Hamiltonian. This contrasts to generic numerical schemes, for which the rate of error growth is superlinear. Asymptotically, the rate of error growth for symplectic schemes is also linear. Hence, Hamiltonian-conserving schemes are competitive with symplectic schemes in this respect. The theory is illustrated with a computation performed on Kepler's problem for the interaction of two bodies.     

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

Preservation of equilibria for symplectic methods applied to Hamiltonian systems

In this paper, the linear stability of symplectic methods for Hamiltonian systems is studied. In par- ticular, three classes of symplectic methods are considered： symplectic Runge-Kutta （SRK） methods, symplectic partitioned Runge-Kutta （SPRK） methods and the composition methods based on SRK or SPRK methods. It is shown that the SRK methods and their compositions preserve the ellipticity of equilibrium points uncondi- tionally, whereas the SPRK methods and their compositions have some restrictions on the time-step.     

A sixth order diagonally implicit symmetric and symplectic Runge-Kutta method for solving Hamiltonian systems

The paper is concerned with construction of symmetric and symplectic Runge-Kutta methods for Hamiltonian systems. Based on the symplectic and symmetrical properties, a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages is presented, the proposed method proved to be P-stable. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientic literature.     

Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems

1.IlltroductiollInmanyareasofphysics,mechanics,etc.,HamiltoniansystemsofODEsplayaveryimportantrole.Suchsystemshavethefollowinggeneralform:where,bydenotingwithOfandimthenullmatrixandtheidentitymatrixofordermarespectively,SimplepropertiesofthematrixJZmarethefollowingones:Inequation(1)AH(~,t)isthegradientofascalarfunctionH(y,t),usuallycalledHamiltonian.InthecasewhereH(y,t)=H(y),thenthevalueofthisfunctionremainsconstantalongt.hesollltion7/(t),t,hatis'*ReceivedFebruaryI3,1995.l)Worksupporte…     

BACKWARD ERROR ANALYSIS OF SYMPLECTIC INTEGRATORS FOR LINEAR SEPARABLE HAMILTONIAN SYSTEMS

Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vg, q = kp are studied. Some new results in connection with P-stability are presented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.     

基于辛格式的深度哈密尔顿神经网络

HNN是一类基于物理先验学习哈密尔顿系统的神经网络.本文通过误差分析解释使用不同积分器作为超参数对HNN的影响.如果我们把网络目标定义为在任意训练集上损失为零的映射,那么传统的积分器无法保证HNN存在网络目标.我们引进反修正方程,并严格证明基于辛格式的HNN具有网络目标,且它与原哈密尔顿量之差依赖于数值格式的精度.数值实验表明,由辛HNN得到的哈密尔顿系统的相流不能精确保持原哈密尔顿量,但保持网络目标;网络目标在训练集、测试集上的损失远小于原哈密尔顿量的损失;在预测问题上辛HNN较非辛HNN具备更强大的泛化能力和更高的精度.因此,辛格式对于HNN是至关重要的.     

A sixth-order dual preserving algorithm for the Camassa-Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.     

CHARACTERIZATIONS OF SYMMETRIC MULTISTEP RUNGE-KUTTA METHODS

Some characterizations for symmetric multistep Runge-Kutta(RK) methods are obtained. Symmetric two-step RK methods with one and two-stages are presented. Numerical examples show that symmetry of multistep RK methods alone is not sufficient for long time integration for reversible Hamiltonian systems. This is an important difference between one-step and multistep symmetric RK methods.