# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Numerical solution of differential and integral equations*

## The Richardson method of high-order accuracy in $t$ for a quasilinear singularly perturbed parabolic reaction-diffusion equation on a strip

**IMM UD RAS (Ekaterinburg)**
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}

*Note. Abstracts are published in author's edition*

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Last update: 06-Jul-2012 (11:52:06)