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

TL;DR

This paper investigates high order nonstandard linear multistep methods, demonstrating they can achieve the same order as standard methods and preserve key qualitative properties under certain conditions, supported by theoretical proofs and numerical experiments.

Contribution

It introduces a nonstandard Taylor series approach and proves conditions for these methods to match standard methods' order and qualitative properties.

Findings

Methods attain the same order as standard counterparts.

Qualitative properties like boundedness are preserved.

Numerical experiments confirm theoretical results.

Abstract

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.