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 Trudy Inst. Mat. i Mekh. UrO RAN, 2014, Volume 20, Number 1, Pages 322–333 (Mi timm1053)

A stable standard difference scheme for a singularly perturbed convection-diffusion equation in the presence of computer perturbations

G. I. Shishkin, L. P. Shishkina

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: A Dirichlet problem approximated by the standard monotone difference scheme on a uniform grid is considered for a singularly perturbed ordinary differential convection-diffusion equation with perturbation parameter $\varepsilon$ ($\varepsilon\in(0,1]$) multiplying the highest-order derivative. Such a scheme does not converge $\varepsilon$-uniformly and, moreover, in the case of its convergence, it is not $\varepsilon$-uniformly well-conditioned and stable to computer perturbations. In this paper, a technique is developed to study solutions of the standard difference scheme in the presence of computer perturbations. Conditions are derived under which the standard finite difference scheme becomes stable to perturbations, necessary and sufficient conditions are obtained for the convergence of computer solutions as the number of grid nodes tends to infinity, and estimates are given for the number of grid nodes (depending on the parameter $\varepsilon$ and computer perturbations $\vartriangle$ defined by the number of computer word digits) for which the error of the numerical solution is smallest.

Keywords: singularly perturbed boundary value problem, convection-diffusion equation, standard difference scheme, uniform grid, maximum norm, conditioning of a difference scheme, perturbed difference scheme, computer perturbations, data perturbations, stable standard difference scheme.

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Document Type: Article
UDC: 519.624

Citation: G. I. Shishkin, L. P. Shishkina, “A stable standard difference scheme for a singularly perturbed convection-diffusion equation in the presence of computer perturbations”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 1, 2014, 322–333

Citation in format AMSBIB
\Bibitem{ShiShi14} \by G.~I.~Shishkin, L.~P.~Shishkina \paper A stable standard difference scheme for a~singularly perturbed convection-diffusion equation in the presence of computer perturbations \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2014 \vol 20 \issue 1 \pages 322--333 \mathnet{http://mi.mathnet.ru/timm1053} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3364215} \elib{http://elibrary.ru/item.asp?id=21258506} 

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This publication is cited in the following articles:
1. G. I. Shishkin, “Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 57:5 (2017), 815–832
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