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

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  • Takacs, B.

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 counterpart — 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.

Suggested Citation

  • Takacs, B., 2026. "An insight on some properties of high order nonstandard linear multistep methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 245(C), pages 337-365.
    • Handle: RePEc:eee:matcom:v:245:y:2026:i:c:p:337-365
    DOI: 10.1016/j.matcom.2026.01.015
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    References listed on IDEAS

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    1. Takács, Bálint & Hadjimichael, Yiannis, 2022. "High order discretization methods for spatial-dependent epidemic models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 198(C), pages 211-236.
    2. Conte, D. & Pagano, G. & Roldán, T., 2026. "High order nonstandard finite-difference methods," Applied Mathematics and Computation, Elsevier, vol. 510(C).
    3. Mokni, Karima & Ch-Chaoui, Mohamed & Mondal, Bapin & Ghosh, Uttam, 2024. "Rich dynamics of a discrete two dimensional predator–prey model using the NSFD scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 992-1018.
    4. Chen-Charpentier, Benito M. & Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2006. "Combined nonstandard numerical methods for ODEs with polynomial right-hand sides," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(1), pages 105-113.
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