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documentstyle[12pt]{article} itle{The Richardson method of high-order accuracy in $t$ for a quasilinear singularly perturbed parabolic reaction-diffusion equation on a strip} author{Pieter W. Hemker, Grigory I. Shishkin, Lidia P. Shishkina} date{} egin{document} maketitle

An initial boundary value problem for a quasilinear parabolic reaction-diffusion equation is considered on a strip. For small values of the perturbation parameter $varepsilon$, the solution of this problem has a singularity, namely a parabolic boundary layer. For such a problem implicit finite difference schemes (base schemes), i.e., a nonlinear scheme (obtained by the classical approximation of the problem) and a non-iterative scheme (such that the unknown function is taken at the previous time level) are developed. Such schemes on piecewise-uniform meshes converge $varepsilon$-uniformly with the order of accuracy no higher than two with respect to the space variables and one with respect to the time variable. In this paper, using Richardson's extrapolation of solutions of the base schemes on piecewise-uniform embedded meshes, we construct discrete solutions that converge $varepsilon$-uniformly at the rate $O(N_1^{-2} ln^2 N_1 +N_2^{-2}+ N_0^{-q})$, $2 le q le 4$, where $N_1$ is the number of mesh intervals on the segment in the $x_1$-direction, $N_2$ is the number of mesh intervals on an unit segment of the $x_2$-axis, and $N_0$ is the number of mesh intervals in the time mesh.

Acknowledgements. This research was supported in part by the Russian Foundation for Basic Research under grant 04--01--00578 and by the Dutch Research Organisation NWO under grant No. 047.016.008. end{document}

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