On modified TDRKN methods for second-order systems of differential equations

ABSTRACT

This paper concerns modified Two-Derivative Runge–Kutta–Nyström (TDRKN) methods for solving second-order initial value problems. Compromised with the deduced order and symmetry criteria, two implicit and forth-order two-stage TDRKN schemes are derived through a mixed collocation approach. Phase and periodic stability features are examined. Numerical experiments are carried out to illustrate the effectiveness and competence of our new methods. Comparisons with existing highly accurate and efficient numerical methods are given.

KEYWORDS:

• Modified Runge–Kutta–Nyström methods
• mixed collocation
• dispersion error
• dissipation error
• trigonometrical fitting

2010 AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L06
• 65L12
• 65L20

Acknowledgements

The authors are grateful to the referees for their valuable suggestions which helped to improve the content and presentation of the paper. Finally, the authors wish to thank Professor Qin Sheng for his invaluable help to enhance the quality and presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Julius O. Ehigie  http://orcid.org/0000-0002-3159-5743

Manman Zou  http://orcid.org/0000-0003-0254-8945

Xinlin Hou  http://orcid.org/0000-0003-0776-2959

Xiong You  http://orcid.org/0000-0002-0343-3878

Additional information

Funding

This work was partially supported by the Natural Science Foundation of China (NSFC) under Grant No. [11171155], the Natural Science Foundation of Jiangsu Province under Grant No. [BK20171370], and State Key Program of Natural Science Foundation of China under Grant No. [31330067].