A generalized second-order positivity-preserving numerical method for non-autonomous dynamical systems with applications

In this work, we propose a generalized, second-order, nonstandard finite difference (NSFD) method for non-autonomous dynamical systems defined by ordinary differential equations (ODEs). 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 a restrictive and indispensable condition required by many existing positivity-preserving, second-order NSFD methods. More precisely, the condition commonly found in literature that the right-hand side functions of ODEs do not vanish at any time step is relaxed. Consequently, 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.