Extended RKN-type methods with minimal dispersion error for perturbed oscillators

Abstract

In this paper, extended Runge–Kutta–Nyström-type methods with minimal dispersion error for the numerical integration of perturbed oscillators are presented, which are based on the order conditions derived by Yang et al. (H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794). The numerical stability and phase properties of the new methods are analyzed. Numerical experiments are reported in comparison with some well-known high quality codes proposed in the scientific literature.

Introduction

This research is concerned with the numerical integration of perturbed oscillators modeled by the initial-value problems of the form

$$\{\begin{array}{c}{y}^{″}(t)+w^{2}y(t)=f(t,y(t)),t\in [t_0,T],\hfill \\ y(t_0)=y_0,y^{\prime }(t_0)=y_0^{\prime },\hfill \end{array}$$

where w is a constant, $f(t,y(t))=\epsilon g(t,y(t))$ with $\epsilon \ll 1$. This kind of problems is frequently encountered in various fields such as celestial mechanics, theoretical physics and chemistry, electronics and so on. They can be integrated by general purpose methods or using other codes adapted to the special structure of the problem (see Refs. [5], [10], [11], [14], [19], [20], [25]). A first good theoretical foundation of this technique was given by Gautschi (see Ref. [12]) and by Lyche (see Ref. [15]). González et al. proposed a new family of Runge–Kutta–Nyström methods for the numerical integration of perturbed oscillators (1) (see Ref. [13]). Using the B-series theory and rooted trees Franco derived the necessary and sufficient order conditions for this class of methods (see Ref. [3]). A further study on the necessary and sufficient order conditions refers to Refs. [29], [30]. Following his approach, Franco constructed a 5(3) pair of explicit adapted Runge–Kutta–Nyström methods (in short notation ARKN) for the particular problems in which the perturbed functions are independent of $y^{\prime }$ (see Ref. [2]); Fang et al. derived a 4(3) pair of explicit ARKN methods for the numerical integration of general perturbed oscillators (see Ref. [9]). Afterwards, using trigonometrical fitting, Yang et al. gave a class of trigonometrically fitted adapted RKN methods (in short notation TFARKN) up to order 5 for the numerical integration of (1) (see Ref. [31]). More recently, using extended tree theory, Yang et al. presented extended RKN (in short notation ERKN) methods based on true flows of both internal stages and updates for the numerical integration of perturbed oscillators (1). However, in that paper, the choice of the nodes $c_i$ is not unique. For a long time it was often believed that the property of zero-dissipation is of primary interest for periodic initial-value problems. On the contrary, rather recently, Van den Houwen and Sommeijer [27], [28] pointed out that the dispersion property is very important for the periodic initial-value problems. Accordingly, they constructed RKN methods with high order dispersion (up to 10) for oscillating problems. Since then, many authors constructed numerical methods with high dispersive order (see Refs. [1], [7], [8], [16], [17], [18], [21], [23], [24]). Therefore, the main purpose of this paper is to investigate the ERKN methods with minimal dispersion error.

This paper is organized as follows. In Section 2, we derive the one-parameter fourth-order ERKN methods for the numerical integration of (1). Using the approach stated in [8], we analyze the stability of the new methods and choose the free parameter such that the dispersion error or the dissipation error is minimal. In Section 3, the numerical experiments are reported. Finally, we are devoted to some conclusive remarks.