Multidimensional ARKN methods for general oscillatory second-order initial value problems
Introduction
Differential equations having oscillatory solutions are of particular interest. Those problems are usually encountered in many fields of the applied sciences and engineering, such as celestial mechanics, theoretical physics, chemistry, electronics, control engineering. A lot of theoretical and numerical researches have been made on the modeling and simulation of these oscillations. Among typical topics is the numerical integration of an oscillatory system associated with an initial value problem of the form where is a positive semi-definite matrix (stiffness matrix, not necessarily symmetric) that implicitly contains the frequencies of the oscillatory problem and , .
The system (1) can be integrated with general purpose methods or using other codes adapted to the special structure of the problem. However, it has become a common understanding that, for the problems with a structure of particular interest, numerical algorithms should respect that structure of the problems. An outstanding advantage of multidimensional ARKN methods for (1) is that their updates take into account the special structure of the system (1) brought by the linear term so that they naturally integrate the multi-frequency and multidimensional unperturbed system exactly. For the work on this topic, we refer the reader to [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].
In [9], the authors constructed the multidimensional ARKN methods for the oscillatory systems whose right-hand side functions are independent of . In this paper we present the analysis of the concrete multidimensional ARKN methods for the oscillatory systems with right-hand side functions depending on both and , based on the order conditions for the ARKN methods derived by Wu et al. [8]. The class of physical problems which fall within its scope is broader in applications. We will discuss the construction of novel multidimensional ARKN methods for the oscillatory system (1) in detail.
The rest of the paper is organized as follows. In Section 2, we restate the basic idea and order conditions of multidimensional ARKN methods for the multi-frequency oscillatory system (1). In Section 3, some multidimensional ARKN methods are proposed based on the order conditions derived in [8]. The stability of the new multidimensional ARKN methods are analyzed. Section 4 gives the numerical experiments. Section 5 is devoted to conclusions and discussions.
Section snippets
Multidimensional ARKN methods and the corresponding order conditions for (1)
In this section, we first introduce the following matrix-valued functions defined in [8]: where is a matrix.
In the recent paper [8], we presented the following matrix-variation-of-constants formula for the exact solution and its derivative for the multi-frequency oscillatory system (1).
Theorem 2.1 If is a symmetric and positive semi-definite matrix and is continuous in (1), then the solution of (1) and its derivative satisfy the following
New multidimensional ARKN methods for oscillatory system (1) with depending on both and
In this section, we construct three explicit ARKN methods for the multi-frequency oscillatory problem (1) and analyze the stability and phase properties of the new methods.
Numerical experiments
In order to show the robustness and efficiency of the novel ARKN methods in comparison with the existing methods in the scientific literature, we use four problems. The methods used for comparison are:
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ARKN3s3: the three-stage ARKN method of order three given in this paper;
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ARKN4s4: the four-stage ARKN method of order four given in this paper;
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ARKN6s5: the six-stage ARKN method of order five given in this paper;
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ARKNVG5: the fifth-order and six-stage ARKN method ARKNVG5 defined by Franco [2]
Conclusions
It is noted that the concrete multidimensional ARKN methods for the general multi-frequency and multidimensional oscillatory second-order initial value problem (1) whose right-hand side functions depend on both and were not constructed in the existing scientific literature, although the order conditions were presented by Wu et al. [8]. In this paper, we make further discussions and analysis on multidimensional ARKN methods for the multi-frequency oscillatory system (1) proposed in paper
Acknowledgments
The authors sincerely thank the anonymous reviewers for their valuable suggestions which considerably improved the paper. The first author would like to thank Numerical Analysis Group of University of Cambridge since the work was partly done when the first author was studying in the group as a joint Ph.D. student under the guidance of Arieh Iserles.
The research was supported in part by the Natural Science Foundation of China under Grant 11271186, by NSFC and RS International Exchanges Project
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