A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

TL;DR

This paper introduces a new second-order nonstandard finite difference method that preserves positivity and stability in one-dimensional autonomous dynamical systems, improving accuracy and extending previous results.

Contribution

A novel second-order NSFD method that maintains positivity and stability, resolving previous conflicts between dynamic consistency and high-order accuracy.

Findings

Method preserves positivity and stability.

Numerical experiments confirm theoretical advantages.

Improves existing NSFD schemes for key equations.

Abstract

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in [Applied Mathematics Letters 112(2021) 106775], [AIP Conference Proceedings 2302(2020) 110003] and [Applied Mathematics Letters 50(2015) 78-82]. Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations; consequently, the NSFD schemes constructed in the earlier work [Journal of Computational and Applied Mathematics 110(1999) 181-185] are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.