Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

ABSTRACT

In this paper, we construct and analyze explicit nonstandard Runge-Kutta (ENRK) methods which have higher accuracy order and preserve two important properties of autonomous dynamical systems, namely, the positivity and linear stability. These methods are based on the classical explicit Runge-Kutta methods, where instead of the usual h in the formulas there stands an appropriately chosen function $\phi (h)$. It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. Some performed numerical simulations confirm the validity of the obtained theoretical results as well as the effectiveness of the proposed method.

Keywords:

• Explicit Runge-Kutta methods
• autonomous dynamical systems
• nonstandard finite difference schemes
• positive finite difference methods
• elementary stable

2010 AMS Subject Classifications:

• 37M
• 65L
• 65Q

Acknowledgments

The authors thank the anonymous referees for useful comments that led to a great improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Quang A Dang http://orcid.org/0000-0001-6386-7347

Manh Tuan Hoang http://orcid.org/0000-0001-6089-3451

Additional information

Funding

This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the [grant number 102.01-2017.306].