Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the line...

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Bibliographic Details
Published in:Journal of difference equations and applications 2024-05, Vol.30 (5), p.577-602
Main Authors: Songolo, Marc E., Bidégaray-Fesquet, Brigitte
Format: Article
Language:English
Subjects:
39-08
exact finite difference schemes
Finite difference method
Mathematical analysis
matrix exponential
Nonlinear systems
nonstandard finite difference schemes
Ordinary differential equations
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Summary:In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley-Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2023.2300726