Two-step extended RKN methods for oscillatory systems☆
Highlights
► Two-step ERKN methods (TSERKN) for oscillatory systems are proposed. ► Order conditions for two-step ERKN methods are presented based on the -series. ► The explicit TSERKN methods are constructed via the order condition derived in this paper. ► The efficiency of the new methods is shown in comparison with the high quality codes 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 where is a symmetric positive semi-definite matrix that implicitly contains the frequencies of the problem, with . 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 so that his ARKN methods (RKN methods adapted to perturbed oscillators [1]) integrate the unperturbed problem 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 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 , fail to equal the values of the exact solution at , 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.
Section snippets
The STS methods and corresponding order conditions
A class of classical two-step hybrid (TS) methods for general second-order initial value problems (2) are given by the following scheme where , and are approximations to , and , respectively. In his fruitful work [18], J.P. Coleman obtained the order conditions for TS methods by using the theory of B2-series. As Coleman said in [18], many other two-step methods, though not
The formulation of the TSERKN methods
Now we turn to the numerical integration of systems of second-order ordinary differential equations (1). We restrict ourselves to the autonomous case of the form for, if contains the time t explicitly, we can extend y by one dimension and rewrite the system equivalently into the following autonomous one
The problem (9) is a special case of the general class of second-order initial value problems of the form
Order conditions
Our next aim is to derive the order conditions for TSERKN methods based on the recently developed EN-trees theory in [9]. Firstly, the reader is referred to that paper for all the definitions and notations. The generalization to the system (9) is direct and the corresponding theory is well established when every in the paper [9] is replaced by the matrix M and every by the matrix V. The concept of branches introduced by [9] and corresponding results are essential for our work in this
Construction of explicit TSERKN methods
In this section we focus on the construction of explicit TSERKN methods with orders four and five.
Stability and phase properties of the new methods
In this section, we are concerned with the stability and phase properties of the new TSERKN integrators. As described in [20], the phase properties of two-step hybrid methods are generally analyzed using the second-order test problem Applying a two-step hybrid method (3) to (44) gets Elimination of the vector Y delivers the recursion where
The phase
Numerical experiments
In this section, in order to show the competence and efficiency of the new methods compared with the well-known methods in scientific literature, we use four model problems whose solutions are known to be oscillatory. The criterion used in the numerical comparisons is the decimal logarithm of the maximum global error (GE) versus the computational effort measured in the number of function evaluations required by each method. The integrators we select for comparison are
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ARKN4s5: the four-stage
Conclusions and discussions
This paper presents a new family of two-step extended RKN-type (TSERKN) methods for oscillatory systems of the form (1). The new methods (18) share the favorable property that they integrate exactly the systems of unperturbed oscillators . Moreover, for the unperturbed oscillators the internal stages of the new methods are also equal to the values of the exact solution at , . Based on the extended Nyström trees theory developed by Yang et al. [9] and -series
Acknowledgements
The authors are grateful to Professor Christian Lubich for his first reading of the manuscript and for his helpful comments. The authors are sincerely thankful to the anonymous referees for their constructive comments and valuable suggestions.
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The research is supported by the Natural Science Foundation of China under Grant 10771099 and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033.