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

2025/05/23 by Takács, Bálint
#65L06 #92D25 #92D30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2505.17606

Abstract

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.

Citations

• High order positivity-preserving numerical methods for a non-local photochemical model
• High-order reliable numerical methods for epidemic models with non-constant recruitment rate
• Second-order nonstandard finite difference schemes for a class of models in bioscience
• Time-Accurate and highly-Stable Explicit operators for stiff\n differential equations
• Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems
• On the use of nonstandard finite difference methods†
• The Formula of Faa Di Bruno
• Total variation diminishing Runge-Kutta schemes

Related

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• Spijker's example and its extension
• Exact optimal values of step-size coefficients for boundedness of linear multistep methods
• A multistep algorithm for ODEs
• High order semi-implicit multistep methods for time dependent partial\n differential equations