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

TL;DR

This paper introduces explicit nonstandard Runge-Kutta methods that improve accuracy while preserving positivity and stability in autonomous dynamical systems, validated through theoretical proofs and numerical simulations.

Contribution

It develops higher-order explicit nonstandard Runge-Kutta methods that maintain key properties of dynamical systems, a novel approach enhancing numerical stability and positivity preservation.

Findings

Methods preserve accuracy order of classical Runge-Kutta methods.

Numerical simulations confirm theoretical stability and positivity.

Constructed methods are applicable to autonomous dynamical systems.

Abstract

In this paper, we construct 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 a function $\varphi (h)$. It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. The numerical simulations confirm the validity of the obtained theoretical results.