High-order closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

doi:10.1016/j.cpc.2007.08.016

Computer Physics Communications

Volume 178, Issue 3, 1 February 2008, Pages 199-207

https://doi.org/10.1016/j.cpc.2007.08.016 Get rights and content

Abstract

In this paper we investigate the connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators. From the literature we can see that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. The well-known open Newton–Cotes differential methods 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 multistep symplectic integrators based on the open Newton–Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, J. Chem. Phys. 107 (1997) 6894]. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. After this we construct 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 integration proceeds.

Introduction

In recent years, the research area of construction of numerical integration methods for ordinary differential equations that preserve qualitative properties of the analytic solution was of great interest. Here we consider Hamilton's equations of motion which are linear in position p and momentum q $\dot{q} = m p,$ $\dot{p} = - m q,$ where m is a constant scalar or matrix. It is well known that Eq. (1) is a an important one in the field of molecular dynamics [3], [5]. In order to preserve the characteristics of the Hamiltonian system in the numerical solution it is necessary to use symplectic integrators. In the recent years work has been done mainly in the construction of one step symplectic integrators (see [11]). In their work Zhu et al. [26] and Chiou and Wu [4] constructed multistep symplectic integrators by writing open Newton–Cotes differential schemes as multilayer symplectic structures.

The last decades much work has been done on exponential fitting and the numerical solution of periodic initial value problems (see [1], [2], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] and references therein).

In this paper we follow the steps described below:

•
We present Closed Newton–Cotes differential methods as multilayer symplectic integrators.
•
We apply the closed Newton–Cotes methods on the Hamiltonian system (1) and we obtain as a result that the Hamiltonian energy of the system remains almost constant as the integration proceeds.
•
We develop trigonometrically-fitted methods.

We note that the aim of this paper is to generate methods that can be used for non-linear differential equations as well as linear ones. The new method developed in this paper has derived from a corresponding classical method and has an extra property without sacrificing any existing properties from the classical one. The new property provides higher efficiency in all problems and especially in problems with oscillating solutions. There is no case that the new method will be less efficient than the corresponding classical since it has the same algebraic order and the extra property.

The construction of the paper is given below. The results about symplectic matrices and schemes are presented in Section 2. In Section 3 we describe closed Newton–Cotes integral rules and differential methods and we develop the new trigonometrically-fitted methods. In Section 4 the conversion of the closed Newton–Cotes differential methods into multilayer symplectic structures is presented. Numerical results are presented in Section 5.

Section snippets

Basic theory on symplectic schemes and numerical methods

The following basic theory on symplectic numerical schemes and symplectic matrices is based on that developed by Zhu et al. [26]. The proposed methods can be used for non-linear differential equations as well as linear ones.

Dividing an interval $[a, b]$ with N points we have $x_{0} = a, x_{n} = x_{0} + n h = b, n = 1, 2, \dots, N .$ We note that x is the independent variable and a and b in the equation for $x_{0}$ (Eq. (2)) are different than the a and b in Eq. (3).

The above division leads to the following discrete scheme: $(\begin{matrix} p_{n + 1} \\ q_{n + 1} \end{matrix}) = M_{n}$

General closed Newton–Cotes formulae

The closed Newton–Cotes integral rules are given by: $\int_{a}^{b} f (x) d x \approx z h \sum_{i = 0}^{k} t_{i} f (x_{i}),$ where $h = \frac{b - a}{N}, x_{i} = a + i h, i = 0, 1, 2, \dots, N .$ The coefficient z as well as the weights $t_{i}$ are given in Table 1.

From Table 1 it is easy to see that the coefficients $t_{i}$ are symmetric, i.e. we have the following relation: $t_{i} = t_{k - i}, i = 0, 1, \dots, \frac{k}{2} .$ Closed Newton–Cotes differential methods were produced from the integral rules. For Table 1 we have the following differential methods: $k = 1 y_{n + 1} - y_{n} = \frac{h}{2} (f_{n + 1} + f_{n}),$ $k = 2 y_{n + 1} - y_{n - 1} = \frac{h}{3} (f_{n - 1} + 4 f_{n} + f_{n + 1}),$ $k = 3 y_{n + 1} -$

Closed Newton–Cotes can be expressed as symplectic integrators

Theorem 1

A discrete scheme of the form $(\begin{matrix} b & - a \\ a & b \end{matrix}) (\begin{matrix} q_{n + 1} \\ p_{n + 1} \end{matrix}) = (\begin{matrix} b & a \\ - a & b \end{matrix}) (\begin{matrix} q_{n} \\ p_{n} \end{matrix})$ is symplectic.

Proof

We rewrite (3) as $(\begin{matrix} q_{n + 1} \\ p_{n + 1} \end{matrix}) = {(\begin{matrix} b & - a \\ a & b \end{matrix})}^{−1} (\begin{matrix} b & a \\ - a & b \end{matrix}) (\begin{matrix} q_{n} \\ p_{n} \end{matrix}) .$

Define $M = {(\begin{matrix} b & - a \\ a & b \end{matrix})}^{−1} (\begin{matrix} b & a \\ - a & b \end{matrix}) = \frac{1}{b^{2} + a^{2}} (\begin{matrix} b^{2} - a^{2} & 2 a b \\ - 2 a b & b^{2} - a^{2} \end{matrix})$ and it can easily be verified that $M^{T} J M = J$ thus the matrix M is symplectic.

In [26] Zhu et al. have proved the symplectic structure of the well-known second-order differential scheme (SOD), $y_{n + 1} - y_{n - 1} = 2 h f_{n},$ $y_{n + 2} - y_{n - 2} = 4 h f_{n},$ $y_{n + 3} - y_{n - 3} = 6 h f_{n} .$

The above methods have been produced by the simplest Open Newton–Cotes integral formula.

Harmonic oscillator

We illustrate the performance of open Newton–Cotes differential methods considering the equations of motion of a harmonic oscillator given by the system of equations: $\dot{q} = p, \dot{p} = - q$ and the initial conditions are given as $q (0) = 1, p (0) = 0 .$ The Hamiltonian (or energy) of this system is $H (t) = \frac{1}{2} (p^{2} (t) + q^{2} (t)) .$

For comparison purposes we use:

•
The seventh order predictor-corrector Adams–Bashforth–Moulton method (which is indicated as method [a]).²

Conclusions

The presentation of the Closed Newton–Cotes differential methods as multilayer symplectic integrators and their application on the Hamiltonian system (1) is presented in this paper. The result from the above investigation was that the Hamiltonian energy of the system remains almost constant as the integration proceeds.

We also developed high order trigonometrically-fitted methods. We applied the new developed methods to linear and nonlinear problems and we compared them with well known

Acknowledgements

The author wishes to thank the anonymous reviewer for his/her careful reading of the manuscript and for the fruitful comments and suggestions.

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¹: Active Member of the European Academy of Sciences and Arts; Corresponding Member of the European Academy of Sciences; Corresponding Member of European Academy of Arts, Sciences and Humanities.

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High-order closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

Abstract

Introduction

Section snippets

Basic theory on symplectic schemes and numerical methods

General closed Newton–Cotes formulae

Closed Newton–Cotes can be expressed as symplectic integrators

Harmonic oscillator

Conclusions

Acknowledgements

New Astronomy

New Astronomy

New Astronomy

New Astronomy

New Astronomy

New Astronomy

New Astronomy

Comput. Phys. Comm.

Mathematical methods of Classical Mechanics

J. Chem. Phys.

Solving Ordinary Differential Equations I

Comput. Lett.

Comput. Lett.

JNAIAM