Modified multi-step Nyström methods for oscillatory general second-order initial value problems

Abstract

Recently, multi-step Nyström methods for the numerical integration of general second-order initial value problems $y^{″}=f(t,y,y^{\prime })$ has been proposed and developed. In this paper, we modify this class of methods to make them adapted to solving numerically oscillatory problems. The new methods have the coefficients depending on v = wh and integrate exactly the problem $y^{″}=-w^{2}y$. General order conditions are given, and two explicit methods with orders three and four, respectively, are constructed. The linear stability analysis of the new methods is considered. Numerical results show that our new methods are more efficient in comparison with other well-known high-quality methods proposed in the scientific literature.

Keywords:

• Modified multi-step Nyström methods
• order conditions
• Nyström-series
• explicit methods
• oscillatory general second-order initial value problems

2010 AMS Subject Classifications:

• 65L05
• 65L06

Acknowledgments

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

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported in part by the Natural Science Foundation of China [grant no: 11401164] and by the Hebei Natural Science Foundation of China [grant no: A2014205136].