Two-step extended RKN methods for oscillatory systems

Abstract

In this paper, two-step extended Runge–Kutta–Nyström-type methods for the numerical integration of perturbed oscillators are presented and studied. The new methods inherit the framework of two-step hybrid methods and are adapted to the special feature of the true flows in both the internal stages and the updates. Based on the EN-trees theory [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], order conditions for the new methods are derived via the ${\mathrm{B}}_{\mathit{BT}}$-series defined on the set BT of branches and the ${\mathrm{B}}_{\mathit{BWT}}$-series defined on the subset BWT of BT. The stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of our new methods in comparison with the well-known high quality methods proposed in the scientific literature.

Introduction

In the last decade, there has been increasing interest in the numerical integration of second-order initial value problems of the form

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

where $M\in {\mathbb{R}}^{m\times m}$ is a symmetric positive semi-definite matrix that implicitly contains the frequencies of the problem, $g(t,y(t))=\epsilon \overline{g}(t,y(t)):\mathbb{R}\times {\mathbb{R}}^m\to {\mathbb{R}}^m$ with $\epsilon \ll 1$. Such problems are frequently encountered in celestial mechanics, theoretical physics, chemistry, electronics, spatial semi-discretizations of wave equations based on the method of lines, and so on. In practice, they can be integrated with general purpose methods or other codes adapted to the special structure of the problem. Generally the adapted methods are more efficient because they make full use of the information transpired from the special structure of (1).

J.M. Franco modified the update of the classical Runge–Kutta–Nyström (RKN) methods to be adapted to the special structure of (1) brought by the term $w^{2}y$ so that his ARKN methods (RKN methods adapted to perturbed oscillators [1]) integrate the unperturbed problem $y^{″}+w^{2}y=0$ exactly. Following [1] are [2], [3], [4], [5], [6]. Afterwards Wu et al. successfully developed the multidimensional ARKN methods by defining ϕ-functions on matrices and gave the corresponding order conditions (see [7] and [8]). However, we have noticed that the internal stages of ARKN methods donʼt take into account the special structure of (1) brought by the term My. For the above reason, H. Yang et al. [9] proposed a new family of extended RKN (ERKN) methods, which take into account the oscillatory feature of the unperturbed oscillators in both the internal stages and the updates. However, the methods are only designed for one-dimensional perturbed oscillators or systems of perturbed oscillators with a diagonal and positive semi-definite matrix M. Following the approach of that paper, Wu et al. [10] formulated a standard form of the multidimensional ERKN integrators for the general system (1) and derived the corresponding order conditions via the theory of the extended Nyström trees in [9].

For general second-order initial value problems of the form

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

many multistep methods and two-step methods have been proposed (see [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]). Generally, two-step methods are known to be more efficient than Runge–Kutta–Nyström methods solving the problem (2) because they need less function evaluations than the latter for achieving the same order. In his fruitful paper [18], J.P. Coleman studied a class of classical two-step hybrid methods for (2) and derived the order conditions based on his B2-series theory.

Recently, H. Van de Vyver [24] revised the updates of the classical two-step hybrid methods and proposed a new family of Scheifele two-step (STS) methods. However, his methods inherit the oscillatory feature of the true flows only in the updates whereas their internal stages have not been revised and then $Y_i$, $(i=1,\dots ,s)$ fail to equal the values of the exact solution $y(t)$ at $t_n+c_{i}h$, even for the unperturbed oscillators. In fact, for adapted TS methods, as foundation of the updates, the internal stages are also important in accuracy.

This paper considers a new family of two-step methods for systems of oscillatory second-order differential equations (1), which take into account the oscillatory feature of the true flows in both the internal stages and the updates.

This paper is organized as follows: in Section 2, we restate the basic idea and order conditions of STS methods. Section 3 proposes the two-step extended RKN (in short notation, TSERKN) methods for the system of oscillatory second-order differential equations (1). Section 4 derives the order conditions of TSERKN methods via B-series theory. With the order conditions achieved in Section 4, three explicit TSERKN methods are proposed in Section 5. In Section 6, the stability and phase properties are considered and the regions of stability or the regions of periodicity, the dispersion errors and the dissipation errors of these methods are given. In Section 7, numerical experiments are accompanied to show the robustness of the new methods. Section 8 is concerned with conclusions.