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\begin{center}
\bf
AN \mbox{\large $\varepsilon$}-UNIFORM DEFFECT-CORRECTION METHOD
FOR A PARABOLIC CONVECTION-DIFFUSION PROBLEM \footnote{
This work was supported in part by the Netherlands Organization
for Scientific Research NWO, dossiernr. 047.003.017, and
by the Russian Foundation for
Basic Research, Grant N 98-01-00362.}
\end{center}

\begin{center}
Pieter W.~Hemker\\ {\small\it CWI, Amsterdam, The Netherlands}\\[1.5ex]
Grigorii I.~Shishkin\\ {\small\it Institute of Mathematics and Mechanics,
Ekaterinburg, Russia}\\[1.5ex]
Lidia P.~Shishkina\\ {\small\it Scientific Research Institute of Heavy
Machine Building, Ekaterinburg, Russia}
\end{center}

\bigskip
On the space-time domain $G=(0,1)\times (0,T]$ we consider
the Dirichlet problem for the singularly perturbed
parabolic equation with convective terms
$$
\begin{array}{c}
\displaystyle \left \{
\varepsilon a(x,t) \frac{{\partial}^2 }{{\partial x}^2}
\displaystyle + b(x,t) \frac{\partial}{\partial x} - c(x,t) -
\displaystyle p(x,t)\frac{\partial}{\partial t} \right \} u(x,t)=
f(x,t), \ \ (x,t) \in G, \\[3ex]
u(x,t) =\varphi(x,t), \quad (x,t)\in \partial G.
\end{array}
\eqno(1)
$$
All the functions in (1) are assumed to be
sufficiently smooth and bounded, moreover,
$0<a_0\leq a(x,t),\ 0<b_0\leq b(x,t), \
0<p_0\leq p(x,t),\ c(x,t)\geq 0$ on $\overline{G}$.
The perturbation parameter $\varepsilon$ may take any small
positive value, say $\varepsilon\in (0,1]$.
As $\varepsilon\to 0$, a boundary layer appears in a
neighbourhood of the left lateral boundary (for $x=0$).

For this class of problems $\varepsilon$-uniform numerical methods,
i.e., methods the accuracy of which is independent of the parameter
$\varepsilon$, are well-known. In particular, it is easy to construct
a piecewise uniform mesh condensing in the boundary layer and
a standard finite difference operator, which give an
$\varepsilon$-uniformly convergent difference scheme.
The accuracy of this base scheme is
$\Oh{N^{-1} \ln^2N + N_0^{-1}}$, where $N+1$ and $N_0+1$ is the
number of nodes in the space and time meshes, respectively.
Therefore, it is of interest to develop schemes with a higher order of
convergence. Here $\varepsilon$-uniformly convergent schemes
with high-order time-accuracy are considered.

To solve problem (1) numerically, we use the base scheme and
the system of additional "corrected" schemes.
The improvement in time-accuracy is achieved by correction of
the right-hand side.
The order of consistency for the new scheme (after
correction) is higher than that for the base scheme. Then we can repeat
such a defect correction in a similar way.
This method with a sequential correction of the local truncation error
allows us to construct $\varepsilon$-uniformly convergent schemes
with an arbitrary high order of time-accuracy
$\Oh{N^{-1} \ln^2N + N_0^{-k}}$, $k\geq 2$.
The efficiency of the new defect-correction scheme in practice is
confirmed by a series of numerical experiments.

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