INTERNATIONAL JOURNAL OF c  2014 Institute for Scientific NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 11, Number 2, Pages 412–426 ON A NUMERICAL TECHNIQUE TO STUDY DIFFERENCE SCHEMES FOR SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION EQUATIONS GRIGORY SHISHKIN, LIDIA SHISHKINA, JOSE LUIS GRACIA, AND CARMELO CLAVERO (Communicated by C. Rodrigo) This paper is dedicated to the 65th birthday of Professor Francisco J. Lisbona Abstract. A new technique to study special difference schemes numerically for a Dirichlet prob- lem 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. 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. 1. Introduction At present, a series of theoretically justified numerical methods convergent ε- uniformly has been elaborated for representative classes of singularly perturbed problems (see, e.g., [3, 5] and the bibliography therein). We also know some “heuris- tic approaches” (they are widely used when solving applied problems) whose jus- tification is rather problematic (see, e.g., [1, 7]). At the same time, nobody knows good experimental methods to study the efficiency of available special grid methods (both theoretical and “heuristic” ones). Thus, the development of experimental methods for numerical study that allow us to reveal the quality of special schemes is an actual problem in the construction of reliable ε-uniformly convergent grid methods for wide classes of singularly perturbed problems. Here we could mention only some interesting numerical researches in [1, 7]. In [1], the two-mesh difference technique has been considered for numerical study of difference schemes on piecewise-uniform grids. This technique has been applied, in particular, to solve a two-dimensional elliptic equation on a rectangle in the case when the convective term includes the derivative along the horizontal axis. The schemes have been considered on meshes with the same number of nodes in both Received by the editors December 3, 2012 and, in revised form, June 14, 2013. 2000 Mathematics Subject Classification. 35B25, 35B45, 65M. 412