In this paper, extended explicit pseudo two-step Runge-Kutta-Nyström (EEPTSRKN) methods for the numerical integration of oscillatory system y″ + My = f(y) are proposed and studied. These methods inherit the framework of explicit pseudo two-step Runge-Kutta-Nyström (EPTSRKN) methods (Cong et al. Comput. Math. Appl. 38, 17–39, 1999) and they are parallel methods. Furthermore, these new methods take into account the special feature of the oscillatory problem so that they integrate exactly unperturbed problem y″ + My = 0. The study of corresponding order conditions shows that an s-stage EEPTSRKN method can obtain maximum step order s + 2 and maximum stage order s + 1. We make a global error analysis and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. Four two-stage methods with step order two, three, three, and four, respectively, are constructed and the corresponding stability regions are given. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.
Runge-Kutta-Nyström methods Order conditions Explicit methods Multi-frequency oscillatory systems Parallel methods
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The authors are grateful to the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.
The research was supported in part by the Natural Science Foundation of China under Grant No: 11401164, by Hebei Natural Science Foundation of China under Grant No: A2014205136, by the Natural Science Foundation of China under Grant No: 11201113 and and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant No: 20121303120001.
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