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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 577–586 (Mi mais524)

Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations

G. I. Shishkin

N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Yekaterinburg 620990, Russia

Abstract: In this paper, for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon^2$, $\varepsilon \in (0,1]$, multiplying the highest-order derivative in the equation, an initial-boundary value Dirichlet problem is considered. For this problem, a standard difference scheme constructed by using monotone grid approximations of the differential problem on uniform grids, is studied in the presence of computer perturbations. Perturbations of grid solutions are studied, which are generated by computer perturbations, i.e., the computations on a computer. The conditions imposed on admissible computer perturbations are obtained under which the accuracy of the perturbed computer solution is the same by order as the solution of an unperturbed difference scheme, i.e., a standard scheme in the absence of perturbations. The schemes of this type with controlled computer perturbations belong to computer difference schemes, also named reliable difference schemes.

Keywords: initial–boundary value problem, singularly perturbed parabolic equation, reaction-diffusion equation, standard difference scheme, uniform grid, computer perturbations, computer difference scheme.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00727_à This research was partially supported by the Russian Foundation for Basic Research under grant ¹ 16-01-00727.

DOI: https://doi.org/10.18255/1818-1015-2016-5-577-586

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UDC: 519.633

Citation: G. I. Shishkin, “Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations”, Model. Anal. Inform. Sist., 23:5 (2016), 577–586

Citation in format AMSBIB
\Bibitem{Shi16} \by G.~I.~Shishkin \paper Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 5 \pages 577--586 \mathnet{http://mi.mathnet.ru/mais524} \crossref{https://doi.org/10.18255/1818-1015-2016-5-577-586} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3569854} \elib{http://elibrary.ru/item.asp?id=27202307}