An insight on some properties of high order nonstandard linear multistep methods

In this paper, nonstandard multistep methods are observed. It is shown that under some (sufficient and necessary) conditions, these methods attain the same order as their standard counterparts - to prove this statement, a nonstandard version of Taylor's series is constructed. The preservation of some qualitative properties (boundedness, a weak form of monotonicity, and the linear combination of the components) is also proven for all step sizes. The methods are applied to a one-dimensional equation and a system of equations, in which the numerical experiments confirm the theoretical results.