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 Kojouharov et al. (2021); Gupta et al. (2020); Wood and Kojouharov (2015). 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 (Mickens, 1999) 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.
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