In this work, we propose a generalized, second-order, nonstandard finite difference (NSFD) method for non-autonomous dynamical systems. The proposed method combines the NSFD framework with a new non-local approximation of the right-hand side function. This method achieves second-order convergence and unconditionally preserves the positivity of solutions for all step sizes. Especially, it avoids the restrictive conditions required by many existing positivity-preserving, second-order NSFD methods. The method is easy to implement and computationally efficient. Numerical experiments, including an improved NSFD scheme for an SIR epidemic model, confirm the theoretical results. Additionally, we demonstrate the method's applicability to nonlinear partial differential equations and boundary value problems with positive solutions, showcasing its versatility in real-world modeling.

翻译：本文提出了一种针对非自治动力系统的广义二阶非标准有限差分（NSFD）方法。该方法将NSFD框架与一种新的右端函数非局部近似相结合，实现了二阶收敛性，并且对任意步长均无条件保持解的正性。特别地，它避免了现有许多二阶保正NSFD方法所需的限制性条件。该方法易于实现且计算高效。数值实验（包括针对SIR传染病模型的一种改进NSFD格式）验证了理论结果。此外，我们展示了该方法在具有正解的非线性偏微分方程和边值问题中的适用性，体现了其在现实世界建模中的广泛适用性。