Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems

The connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators is studied in this paper. Several one-step symplectic integrators have been obtained based on symplectic geometry, as is shown in the literature. However, the study of multi-step symplectic integrators is very limited. The well-known open Newton–Cotes differential methods are presented as multilayer symplectic integrators by Zhu et al. [W. Zhu, X. Zhao, Y. Tang, Journal of Chem. Phys. 104 (1996), 2275]. The construction of multi-step symplectic integrators based on the open Newton–Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, Journal of Chemical Physics 107 (1997), 6894]. The closed Newton–Cotes formulae are studied in this paper and presented as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton’s equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as the integration proceeds. Finally we apply the new developed methods to an orbital problem in order to show the efficiency of this new methodology.     

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.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

Symplectic exponentially-fitted four-stage Runge–Kutta methods of the Gauss type

The construction of symmetric and symplectic exponentially-fitted Runge–Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions deserves a lot of interest. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators of eighth-order are studied and constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (2004). Numerical experiments for some oscillatory problems are presented.     

Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods

Summary We consider the question of whether multistep methods inherit in some sense quadratic first integrals possessed by the differential system being integrated. We also investigate whether, in the integration of Hamiltonian systems, multistep methods conserve the symplectic structure of the phase space.     

Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric Integration of Ordinary Differential Equations on Manifolds

This article illustrates how classical integration methods for differential equations on manifolds can be modified in order to preserve certain geometric properties of the exact flow. Projection methods as well as integrators based on local coordinates are considered. The central ideas of this article have been presented at the 40th anniversary meeting of the journal BIT.This revised version was published online in October 2005 with corrections to the Cover Date.     

Higher-order Newton–Cotes rules with end corrections

We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.     

Structure preserving integrators for non-linear structural dynamics

A well known and major drawback of standard time integration schemes in the field of non-linear elastodynamics is their unstable behavior in the case of stiff material behaviour. Even second order accurate implicit time integration schemes are unable to resolve the problem under consideration effectively. To remedy this drawback, structure preserving integrators have been developed. Therefore, the goal of this paper is to compare recently developed integrators. In particular, an energy and momentum conserving scheme, based on a publication by Betsch & Steinmann [1], as well as a symplectic variational integrator, proposed by Lew et al. [4] and Wendlandt & Marsden [3], based on a mid-point evaluation of the discrete Lagrangian, are presented. Two representative numerical examples will outline the characteristics of the different approaches. In particular, a stiff non-linear spring pendulum and a finite element model of non-linear structural dynamics are considered. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analogue of Newton–Cotes formulas for numerical integration of functions with a boundary-layer component

The numerical integration of functions with a boundary-layer component whose derivatives are not uniformly bounded is investigated. The Newton–Cotes formulas as applied to such functions can lead to significant errors. An analogue of Newton–Cotes formulas that is exact for the boundary-layer component is constructed. For the resulting formula, an error estimate that is uniform with respect to the boundary-layer component and its derivatives is obtained. Numerical results that agree with the error estimates are presented.