Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems

^** Email: David.Cohen{at}math.unige.ch. Present address: Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany (cohen{at}na.uni-tuebingen.de) Modulated Fourier expansion is used to show long-time near-conservationof the total and oscillatory energies of numerical methods forHamiltonian systems with highly oscillatory solutions. The numericalmethods considered are an extension of the trigonometric methods.A brief discussion of conservation properties in the continuousproblem and in the multi-frequency case is also given.     

High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.     

Modulated Fourier Expansions of Highly Oscillatory Differential Equations

Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi–Pasta–Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discretizations.     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

基于修正的Magnus方法的高振荡动力系统的数值积分方法

基于建立于一般线性动力系统上的Magnus数值积分方法,针对随时间而高频率振荡的二阶动力系统,给出了有效的修正Magnus数值积分算法．首先,将二阶动力系统重新表示为一阶系统的形式,通过引进新变量进行参考坐标变换,使动力系统的高振荡性质保留在新形式内;进而基于局部线性化技术用修正的Magnus方法求解新形式下的系统方程;最后,通过一系列数值实验说明了文中方法的有效性．     

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.     

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.     

线性矩阵哈密顿系统的振动性

研究了线性矩阵 Hamilton系统X′=A( t) X + B( t) YY′=C( t) X -A*( t) Y　　 t≥ 0的振动性 .其中 A( t) ,B( t) ,C( t) ,X,Y为实 n× n矩阵值函数 ,B,C为对称矩阵 ,B正定 .借助于正线性泛函 ,采用加权平均法 ,得到了该系统的非平凡预备解的振动性 .这些结果推广、改进了许多已知的结果     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

ON ENERGY CONSERVATION BY TRIGONOMETRIC INTEGRATORS IN THE LINEAR CASE WITH APPLICATION TO WAVE EQUATIONS

Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered. Under a condition of Hairer & Lubich on the filter functions in the method, a modified energy is derived that is exactly preserved by trigonometric integrators. This implies and extends a known result on all-time near-conservation of energy. The extension can be applied to linear wave equations.     

Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes

In this paper we consider an analytic Hamiltonian system differing from an integrable system by a small perturbation of order . The corresponding unperturbed integrable system is degenerate with proper and limit degeneracy: all variables, except two, are at rest and there is an elliptic singular point in the plane of these two variables. It is shown that by an analytic symplectic change of the variable, which is -close to the identity substitution, the Hamiltonian can be reduced to a form differing only by exponentially small ( ) terms from the Hamiltonian possessing the following properties: all variables, except two, change slowly at a rate of order and for the two remaining variables the origin is the point of equilibrium; moreover, the Hamiltonian depends only on the action of the system linearized about this equilibrium.     

Lyapunov Center Theorem of Infinite Dimensional Hamiltonian Systems

In this paper we reformulate a Lyapunov center theorem of infinite dimensional Hamiltonian systems arising from PDEs. The proof is based on a modified KAM iteration for periodic case.     

The Isoenergetic KAM-Type Theorem at Resonant Case for Nearly Integrable Hamiltonian Systems

In this paper, we study the persistence of resonant invariant tori on energy surfaces for nearly integrable Hamiltonian systems under the usual R$\ddot{u}$ssmann nondegenerate condition. By a quasilinear iterative scheme, we prove the following things: (1) The majority of resonant tori on a given energy surface will be persisted under R$\ddot{u}$ssmann nondegenerate condition. (2) The maximal number about the preserved frequency components on a perturbed torus is characterized by the smaller of the maximal rank of the Hessian matrices of the unperturbed system and the nondegeneracy of resonance. (3) If unperturbed systems admit subisoenergetic nondegeneracy on an energy surface, then the majority of the unperturbed resonant tori on the energy surface will be persisted and give rise to a family of perturbed tori with the same energy, whose frequency ratios among respective ''nondegenerate'' components are preserved.     

Spectral Semi-discretisations of Weakly Non-linear Wave Equations over Long Times

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansions. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems

Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians developed recently in Ghoussoub (2007a Ghoussoub , N. ( 2007a ). Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions . AIHP-Analyse Non Linéaire 24 : 171 – 205 . [Google Scholar] b Ghoussoub , N. ( 2007b ). Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions . Comm. Pure & Applied Math. 60 ( 5 ): 619 – 653 .[Crossref], [Web of Science ®] , [Google Scholar] c Ghoussoub , N. ( 2007c ). Selfdual partial differential systems and their variational principals . Submitted for publication . [Google Scholar]).     

哈密顿系统的低维环面的保存性

本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.     

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．     

Monotonicity of the ratio of two Abelian integrals for a class of symmetric hyperelliptic Hamiltonian systems

In this paper we study the monotonicity of the ratio of two hyperelliptic Abelian integrals $I_0(h)=\oint_{\Gamma_h}ydx$ and $I_1(h)=\oint_{\Gamma_h}xydx$ for which $\Gamma_h$ is a continuous family of periodic orbits of a Newtonian system with Hamiltonian function of the form $H(x,y)=\frac{1}{2}{y^2}\pm \Psi(x)$, where $\Psi$ is an arbitrary even function of degree six.