Int. J. Numer. Anal. Mod., 11 (2014), pp. 412-426.


On a numerical technique to study difference schemes for singularly perturbed parabolic reaction-diffusion equations

Grigory Shishkin 1, Lidia Shishkina 1, Jose Luis Gracia 1, Carmelo Clavero 1

1 Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, GSP-384, Russia and School of Engineering and Architecture, Department of Applied Mathematics, University of Zaragoza, 50018 Zaragoza, SPAIN

Received by the editors December 3, 2012 and, in revised form, June 14, 2013

Abstract

A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε ; ε ∈ (0, 1]. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme converges ε-uniformly in the maximum norm at the rate O(N^{-2} ln^2 N + N^{-1}_0) as N, N_0 → ∞, where N + 1 and N_0 + 1 are the numbers of nodes in the spatial and time meshes, respectively; for ε ≥ m ln^{-1} N the scheme converges at the rate O(N^{-2} + N^{-1}_0). In this paper we elaborate a new approach based on the consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both variables x and t), and also fractional errors (in x and in t) generated in the approximation of differential derivatives by grid derivatives. The regularized total errors agree well with known theoretical estimates for actual errors and their convergence rate orders. It is also shown that a “standard” approach based on the “fine grid technique” turns out inefficient for numerical study of difference schemes because this technique brings to large errors already when estimating the total actual error.

AMS subject classifications: 35B25, 35B45, 65M
Key words: parabolic reaction-diffusion equation, perturbation parameter, boundary layer, difference scheme, piecewise-uniform grids, ε-uniform convergence, numerical experiments, total error, fractional errors, regularized errors.

Email: shishkin@imm.uran.ru and jlgracia@unizar.es
 

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