Multi-step Nyström methods for general second-order initial value problems y″(t) = f(t,y(t), y′(t))

ABSTRACT

A class of multi-step Nyström methods for solving numerically general second-order initial value problems $y^{\mathrm{\prime \prime }}=f(t,y,y^{\mathrm{\prime }})$ are proposed and developed. The new methods inherit the main frameworks of classical Nyström methods and make use of the information of previous steps. Order conditions are deduced by using the theory of Nyström-series based on the set of Nyström-trees, and two explicit methods with convergence orders three and four, respectively, are constructed. The linear stability of the new methods is analysed. Numerical results show that our new methods are more efficient in comparison with Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.

KEYWORDS:

• Multi-step Nyström methods
• order conditions
• Nyström-series
• explicit methods
• general second-order initial value problems

2010 AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L06

Acknowledgments

The authors are grateful to the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported in part by the Natural Science Foundation of China [Grant No: 11401164], by Hebei Natural Science Foundation of China [Grant No: A2014205136], by the Natural Science Foundation of China [Grant No: 11201113] and by the Specialized Research Foundation for the Doctoral Program of Higher Education [Grant No: 20121303120001].