一类通用动力系统的简单二阶非标准数值方法及其应用
A simple second-order nonstandard numerical method for a general class of dynamical systems and its applications
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中文总结 AI 辅助
本文提出一种二阶非标准有限差分方法，可保留动力系统的解正性与平衡点渐近稳定性，将其应用于两阶段结构化种群模型，数值实验显示该方法优于显式梯形法。
中文摘要 AI 辅助
本研究考虑一类连续时间自治动力系统，其可建模现实场景中各类重要现象与过程。我们构造了一种二阶非标准有限差分（NSFD）方法，该方法在所有有限步长下同时保留动力系统的两个核心性质：解的正性、平衡点集合及其渐近稳定性。此NSFD方法基于非标准分母函数的合理选择与右端函数的加权离散化构建；在易于验证的条件下，分母函数保证二阶收敛性，权重确保动态一致性。利用右端函数的特定结构，我们采用简单离散化替代了以往研究中常用的非局部离散化方法，简化了所提NSFD方法的构造，尤其使其渐近稳定性分析更简便。作为示例及重要应用，我们将所构造的二阶NSFD方法应用于著名的带补充的两阶段结构化种群模型，由此推导出该两阶段结构化种群模型对应的简单二阶NSFD格式，改进了以往研究中构造的一阶NSFD格式。数值实验表明，该二阶NSFD格式优于标准二阶数值方法——显式梯形法。所提NSFD方法简便，可应用于理论及应用中出现的各类通用动力系统模型，还可轻易与Richardson外推技术结合以提升精度。
英文摘要
In this work, we consider a class of continuous-time autonomous dynamical systems that model various important phenomena and processes encountered in real-world situations. We construct a second-order nonstandard finite difference (NSFD) method that simultaneously preserves two essential properties of the dynamical systems for all finite step sizes, namely the positivity of the solutions, the set of equilibrium points and their asymptotic stability. This NSFD method is constructed based on an appropriate choice of nonstandard denominator functions and a weighted discretization of the right-hand side functions. Under easily-verified conditions, the denominator functions guarantee second-order convergence, whereas the weights ensure the dynamic consistency. By taking advantage of the specific structure of the right-hand side functions, a simple discretization is utilized instead of the nonlocal discretization approaches commonly used in previous works. This simplifies the construction of the proposed NSFD method and, in particular, makes its asymptotic stability analysis easier. As an illustration and an important application, we apply the constructed second-order NSFD method to a well-known two-stage structured species model with recruitment. Consequently, a simple second-order NSFD scheme for the considered two-stage structured species model is derived, improving upon a first-order NSFD scheme constructed in a previous work. Numerical experiments demonstrate the advantages of the second-order NSFD scheme over a standard second-order numerical method, namely, the explicit trapezoidal method. The proposed NSFD method is simple and can be applied to a broad class of dynamical system models arising in both theory and applications. Moreover, it can be readily combined with the Richardson extrapolation technique to improve its accuracy.
