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\title{The Numerical Solution of a Neumann Problem\\
 for Parabolic Singular Perturbed Equations\\
 with High-Order Time Accuracy}
\thanks{This research was supported in part by the Dutch
Research Organisation NWO under grant N 047.003.017 and
by the Russian Foundation for Basic
Research under grant N 98-01-00362.}}
\author{P.W.~Hemker\thanks{ CWI, Amsterdam, The Netherlands.}
 \and
\ G.I.~Shishkin\thanks{ Institute of Mathematics and Mechanics,
   Ural Branch of the Russian Academy of Science, Ekaterinburg, Russia.}
 \and
\ L.P.~Shishkina\thanks{ Scientific Research Institute of Heavy
   Machine Building, Ekaterinburg, Russia.}
}
\maketitle

 We study the discrete approximation of a Neumann problem
on an interval for a singularly perturbed parabolic PDE.
For this boundary value problem we construct a special piecewise-uniform
mesh on which the discretisation, based on the classical finite difference
approximation, converges $\epsilon$-uniformly with the order
${\cal O} (N^{-2}\ln^2 N+K^{-1})$, where, respectively,
$N$ and $K$ are the number of intervals in the space  and the time mesh.
With such discretisations we construct schemes of  high order
accuracy in the time.
To obtain the better accuracy, we use a defect correction technique.
Auxiliary discrete problems on the same time-mesh are introduced
to correct the low order difference approximations.
To validate the theoretical results, some numerical results
for the new schemes are presented.

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