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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

求解扰动振子的三角拟合RKN方法及其高阶

组合方法

王斌，吴新元

南京大学数学系，南京　210093

摘要：基于Tocino等的修正Runge-Kutta-Nystr¨om (RKN)方法 ([Math. Comput. Model., (42)

2005])，本文探讨求解扰动振子y

+ ω

y(t) = f(y(t)) 高效率的显式三角拟合RKN方法。将组

合方法技术应用于三角拟合RKN方法，本文导出了一些高阶方法。与文献中已有的经典方法

相比，数值试验结果表明本文的高阶显式三角拟合RKN方法更加有效。

关键词：三角拟合RKN方法，辛条件，扰动振子，组合方法

中图分类号： O241.81

Higher-order trigonometrically ﬁtted RKN

schemes via composition methods for solving

perturbed oscillators

WANG Bin , WU Xin-Yuan

Department of Mathematics, Nanjing University, Nanjing 210093

Abstract: This paper devotes to exploring eﬃcient explicit trigonometrically ﬁtted

Runge-Kutta-Nystr¨om methods for the perturbed oscillators y

+ ω

y(t) = f(y(t)) based on

the modiﬁed Runge-Kutta-Nystr¨om methods proposed by Tocino et al. [Math. Comput.

Model., (42) 2005]. Based on composition methods, some eﬃcient explicit trigonometrically

ﬁtted RKN methods are proposed. Numerical results demonstrate that our higher-order

explicit trigonometrically ﬁtted Runge-Kutta-Nystr¨om methods are more eﬃcient than the

well-known methods in the scientiﬁc literature.

Key words: trigonometrically ﬁtted RKN methods, symplecticity conditions, perturbed

oscillators, composition methods.

基金项目： Specialized Research Foundation for the Doctoral Program of Higher Education (20100091110033)，

Natural Science Foundation of China (11271186)，985 Project at Nanjing University (9112020301), University Post-

graduate Research and Innovation Project of Jiangsu Province 2012 (CXZZ12

−

0028).

作者简介： Wang Bin（1986-），male，PHD student，major research direction：Numerical methods for ordinary

diﬀerential equations. Correspondence author：Wu Xin-Yuan（1953-），male，professor，major research direction：

Numerical methods for ordinary diﬀerential equations.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

In the past over two decades, symplectic methods for Hamiltonian systems have formed

a very important category in the numerical solution of ordinary diﬀerential equations. For

diﬀerential equations with particular structures, it has become a common practice to require

numerical algorithms to adapt the special structure of the problems and preserve as much as

possible the qualitative behavior of the true solutions. Recent survey of structure-preserving

algorithms for ordinary diﬀerential equations, we refer to [1, 2].

Oscillatory systems often arise in diﬀerent ﬁelds of applied sciences such as the Kepler

problem, the outer solar system, the H´enon-Heiles model and problems in molecular dynamics,

etc (see [1, 3] for example). Several of these problems can be expressed in the following system

of second order diﬀerential equations:

+ ω

y(t) = f(y(t)), y(t

) = y

, y

) = y

, (1)

where ω is the main frequency, y : R → R

, and f(y) is the perturbing force with the form

f(y) = −∇U(y). The system (1) is in fact a Hamiltonian system with Hamiltonian H(y

, y) =

y + U(y).

In order to deal with the problem (1), some novel approaches to exploring numerical

methods have been proposed and we refer to [4, 5, 6, 7, 8, 9, 10, 11, 12] for example. For problem

(1), Tocino et al. propose a scheme of modiﬁed RKN methods and discuss the symplecticity

conditions in [7]. In this paper, we apply trigonometrical ﬁtting technique to the modiﬁed

RKN methods to achieve the so called trigonometrically ﬁtted RKN methods for problem

(1). The corresponding symplecticity conditions for trigonometrically ﬁtted RKN methods

are obtained by using the results given in [7]. Based on the symplecticity conditions and

composition methods, we propose some practical explicit symplectic trigonometrically ﬁtted

RKN methods. Three numerical experiments are carried out to demonstrate the eﬃciency of

these new methods.

1 Trigonometrically ﬁtted RKN methods

For the numerical solution of problem (1), modiﬁed RKN methods of the following form

are introduced in [7]:

= C

+ hD

+ h

j=1

f(Y

), i = 1, 2, · · · , s,

n+1

= Cy

+ hDy

+ h

i=1

f(Y

n+1

= Ay

+ h

i=1

f(Y

(2)

- 2 -

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