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.

翻译：在这项工作中,引入并分析了一种新颖的二级非标准定点差异法,该方法同时保存了一维自主动态系统的正反常和局部无线稳定性,其基础是不同方程式与非标准分母功能的右侧功能的非局部新近似值,其成果不仅解决了非标准定点法方法动态一致性和高端准确性之间的矛盾,而且改进并扩展了最近在[应用数学信112(2021)106775)、[AIP会议记录2302(202020)11003]和[应用数学信50(2015)78-82]中公布的一些众所周知的结果。此外,作为一个简单但重要的应用,我们应用了已构建的NSDFD方法来解决物流、正弦、立方和单方方程式之间的矛盾;因此,在早期工作中构建的NSDFD计划[比较和应用数学杂志(1999年)110-181-185]得到显著改进。最后,我们报告了一些数字实验,以支持和说明SFDA的理论主张,并说明了其优势。