A SYMPLECTIC ALGORITHM FOR DYNAMICS OF RIGID BODY

For the dynamics of a rigid body with a fixed point based on the quaternion and the corresponding generalized momenta, a displacement-based symplectic integration scheme for differential-algebraic equations is proposed and applied to the Lagrange＇s equations based on dependent generalized momenta. Numerical experiments show that the algorithm possesses such characters as high precision and preserving system invariants. More importantly, the generalized momenta based Lagrange＇s equations show unique advantages over the traditional Lagrange＇s equations in symplectic integrations.

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Biicklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic con- servative perturbation method for linear time-varying Hamil- tonian system. Firstly, the dynamic equation of Hamilto- nian system is gradually changed into a high order pertur- bation equation, which is solved approximately by resolv- ing the Hamiltonian coefficient matrix into a ＂major compo- nent＂ and a ＂high order small quantity＂ and using perturba- tion transformation technique, then the solution to the orig- inal equation of Hamiltonian system is determined through a series of inverse transform. Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes, the transfer matrix is a sym- plectic matrix; furthermore, the exponential matrices can be calculated accurately by the precise time integration method, so the method presented in this paper has fine accuracy, ef- ficiency and stability. The examples show that the proposed method can also give good results even though a large time step is selected, and with the increase of the perturbation or- der, the perturbation solutions tend to exact solutions rapidly.

An Integrable Symplectic Map Related to Discrete Nonlinear Schrdinger Equation

The method of nonlinearization of spectral problem is developed and applied to the discrete nonlinear Schr6dinger （DNLS） equation which is a reduction of the Ablowitz-Ladik equation with a reality condition. A new integable symplectic map is obtained and its integrable properties such as the Lax representation, r-matrix, and invariants are established.

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respectively,are considered in this paper.Stochastic Runge-Kutta(SRK)methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplectic property are given.Based on the weak/strong order and symplectic conditions,some effective schemes are derived.In particular,using the algebraic computation,we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise,and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise,respectively.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

SYMPLECTIC SCHEMES FOR QUASILINEAR WAVE EQUATIONS OF KLEIN-GORDON AND SINE-GORDON TYPE

Presents a class of finite difference methods of first and second order accuracy for the computation of solutions to the quasilinear wave equations. Overview of general symplectic schemes and generating functional; Numerical schemes for quasilinear wave equations; Numerical results.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied.An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order.In general the proposed symplectic schemes are fully implicit,and they become computationally expensive for mean square orders greater than two.However,for stochastic Hamiltonian systems preserving Hamiltonian functions,the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes.A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Symplectic orbit propagation based on Deprit’s radial intermediary

The constantly challenging requirements for orbit prediction have opened the need for better onboard propagation tools.Runge-Kutta(RK)integrators have been widely used for this purpose;however RK integrators are not symplectic,which means that RK integrators may lead to incorrect global behavior and degraded accuracy.Emanating from Deprit’s radial intermediary,obtained by the elimination of the parallax transformation,we present the development of symplectic integrators of different orders for spacecraft orbit propagation.Through a set of numerical simulations,it is shown that these integrators are more accurate and substantially faster than Runge-Kutta-based methods.Moreover,it is also shown that the proposed integrators are more accurate than analytic propagation algorithms based on Deprit’s radial intermediary solution,and even other previously-developed symplectic integrators.

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.

VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs

Discusses the symplectic integration of separable Hamiltonian ordinary and partial differential equations (PDE). Results of a von Neumann analysis; Spectra of linearized Hamiltonian PDEs.

NEW SYMPLECTIC MAPS: INTEGRABILITY AND LAX REPRESENTATION

New family of integrable symplectic maps are reduced from the Toda hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions.Their integrability and Lax representation are deduced systematically from the discrete zero curvature representation of the Toda hierarchy. Also a discrete zero curvature representation for the Toda hierarchy with sources is presented.

GeometricNumerical Integration for Peakon b-Family Equations

In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.

Symplectic Euler Method for Nonlinear High Order Schr¨odinger Equation with a Trapped Term

In this paper,we establish a family of symplectic integrators for a class of high order Schrodinger 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 with 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.

Factorization of the Toda Hierarchy and Poisson Structure for Symplectic Maps

It is shown that each lattice equation in the Toda hierarchy can be factored by an integrable symplectic map and a finite dimensional integrable Hamiltonian system via higher order constraint relating the potential and square eigenfunctions. The classical Poisson structure and r matrix for the constrained flows are presented.

Tuning Symplectic Integrators is Easy and Worthwhile

Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized al-gorithms and from choosing the best ordering of terms.The cost in programming and execution time is minimal,while the performance improvements can be large.In this note we report the influence of term ordering for random systems and for two systems from celestial mechanics that describe particle paths near black holes,quantifying its significance for both optimized and unoptimized methods.We also present a method for the computation of solutions of integrable monomial Hamiltonians that minimizes roundoff error and allows the effective use of compensation summation.

An Integrable Symplectic Map of a Differential-Difference Hierarchy

By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense.

Conjugacy Classes of Torsion in 4×4 Integral Symplectic Group

A complete list of representatives of conjugacy classes of torsion in 4×4 integral symplectic group is given in this paper. There are 55 distinct such classes and each torsion element has order of 2, 3, 4, 5, 6, 8, 10 and 12.

BACKWARD ERROR ANALYSIS OF SYMPLECTIC INTEGRATORS FOR LINEAR SEPARABLE HAMILTONIAN SYSTEMS

Presents a study that analyzed the symplecticness, stability and asymptotic of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nystr ? m methods applied to linear Hamiltonian systems. Numerical representation of the problem; Results in connection to P-stability; Details of the application of backward error analysis in the study.

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi＇s solution of the Hamilton- Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators We also illustrate these systematic methods for constructing variational integrators with numerical examples.

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 symplectie map and a completely integrable tinite-dimensionai Hamiltonian system.