High order nonstandard finite-difference methods

Nonstandard finite difference (NSFD) methods have been considered to overcome some issues of standard methods, particularly when the numerical solution must preserve important properties of the exact solution. These issues increase for high order methods. In this paper we first derive a general procedure to obtain unconditionally positive second order NSFD methods. Furthermore, by suitably adding some parameters ai within these schemes, we show that it is still possible to get positivity, and also to preserve other qualitative properties of the exact solution. In fact, for each particular problem we can get optimal values of ai that guarantee positivity, elementary stability and the minimization of the local truncation error, being possible to achieve also third order nonstandard schemes, which are not present in the literature. As an example of use, we employ the developed theory to derive positive and elementary stable NSFD methods of order one, two and three for a predator-prey model, showing their advantages over other nonstandard methods from the literature.