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

In this paper, nonstandard multistep methods are considered. 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, the linear combination of the components , and a property similar to monotonicity) 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. • Under some conditions, nonstandard linear multistep methods have the same order as their standard counterparts. • These nonstandard methods also preserve some vital qualitative properties (boundedness, sum of components) of the original, continuous model for any choice of step size. • An algorithm is presented for constructing such nonstandard methods, and it is applied to two examples.