Nonstandard finite difference method by nonlocal approximation

Two types of monotonic properties of solutions of differential equations are discussed and general finite difference schemes, which are stable with respect to these properties are investigated. Apart from being elementary stable, these schemes are also shown to preserve qualitative properties of nonhyperbolic fixed points of the differential equations. From the practical point of view, a systematic procedure based on nonlocal approximation, is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.