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.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.     

Equivariant constrained symplectic integration

Summary We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum.     

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

     

Symplectic group actions and covering spaces

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied.     

An energy-momentum conserving scheme for Hamiltonian wave equation based on multiquadric trigonometric quasi-interpolation

Based on multiquadric trigonometric quasi-interpolation, the paper proposes a meshless symplectic scheme for Hamiltonian wave equation with periodic boundary conditions. The scheme first discretizes the equation in space using an iterated derivative approximation method based on multiquadric trigonometric quasi-interpolation and then in time with an appropriate symplectic scheme. This in turn yields a finite-dimensional semi-discrete Hamiltonian system whose energy and momentum (approximations of the continuous ones) are invariant with respect to time. The key feature of the scheme is that it conserves both the energy and momentum of the Hamiltonian system for both uniform and scattered centers, while classical energy-momentum conserving schemes are only for uniform centers. Numerical examples provided at the end of the paper show that the scheme is efficient and easy to implement.     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

Backward error analysis for multi-symplectic integration methods

Summary. A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations obtained through backward error analysis, and using symplectic integration for Hamiltonian ODEs provides more incite into the modified equations. In this paper, the ideas of symplectic integration are extended to Hamiltonian PDEs, and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize, but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. These results are also applied to the nonlinear wave equation. Mathematics Subject Classification (1991):65M06, 65P10, 37K05     

A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems

In this paper we consider the necessary conditions of conservation laws of symplectic difference schemes for Hamiltonian systems and give an example which shows that there does not exist any centered symplectic difference scheme which preserves all Hamiltonian energy.     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Numerical integration of Hamiltonian problems by G-symplectic methods

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.     

Hamilton–Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties

In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Störmer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.     

Geometric integration of the paraxial equation

Evolution of solitary waves in photovoltaic-photorefractive crystal satisfy the paraxial equation. The paraxial equation is transformed into the symplectic structure of the infinite dimensional Hamiltonian system. The symplectic structure of the paraxial equation is discretizated by the symplectic method. The corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation. The symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation. Evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated.     

On Hamiltonian rational transfer functions

A sequel to a previous paper, Fuhrmann [1983], this paper contains a detailed study of rational transfer functions having Hamiltonian symmetry. A detailed realization procedure is described that reflects this symmetry. As a corollary the Williamson canonical form for Hamiltonian maps in symplectic spaces is derived.     

The linear symmetries of Hill’s lunar problem

Archiv der Mathematik - A symmetry of a Hamiltonian system is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. For the planar and spatial Hill lunar problem, four...     

Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

The symplectic reduced spaces of a Poisson actionLes espaces réduits symplectiques d'une action de Poisson

During the last thirty years, symplectic or Marsden–Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this Note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. In other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted upon canonically by a Lie group. To cite this article: J.-P. Ortega, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 999–1004.     

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.