High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

In this work, we introduce a simple and efficient approach for constructing dynamically consistent and high-order nonstandard finite difference (NSFD) methods for a class of nonlinear dynamical systems. These high-order NSFD methods are derived from a novel weighted non-local approximation of the right-hand side functions and the renormalization of nonstandard denominator functions. It is proved by rigorous mathematical analysis that the NSFD methods can be accurate of arbitrary order and preserve two essential mathematical features including the positivity and asymptotic stability of the dynamical systems under consideration for all finite values of the step size. In the constructed NSFD methods, the weighted non-local approximation guarantees the dynamic consistency and the renormalized nonstandard denominator functions ensure that the proposed NSFD methods are accurate of arbitrary order. The obtained results resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD schemes and hence, improve several well-known results focusing on NSFD methods for differential equations. Finally, illustrative numerical examples are reported to support the theoretical findings and to show advantages of the high-order NSFD methods.