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\begin{document}
\title{Numerical Analysis in Singularly Perturbed Boundary Value
Problems Modelling Heat Transfer Processes\thanks{This work
was supported by the Russian Foundation of Basic Research under Grant
N 95-01-00039a}}
\author{V.L. Kolmogorov\inst{1}, G.I. Shishkin\inst{2} and
L.P.Shishkina\inst{3}}
\institute{Institute of Engineering Science, Ural Branch of Russian
Academy of Sciences, GSP-207 620219, Ekaterinburg, Russia
\and
Institute of Mathematics and Mechanics,
Ural Branch of Russian Academy of Sciences, GSP-384
620219 Ekaterinburg, Russia
\and
Scientific Research Institute of Heavy Machine Building, Ekaterinburg,
Russia
}
\maketitle
\begin{abstract}
We construct a finite difference method for
boundary value problems modelling heat and mass transfer for fast-running
processes. The dimensionless form of the equation in these problems is
singularly perturbed, i.e., the highest derivatives are
multiplied by a parameter $\eps^2$ which can take any values from the
interval (0,1]. The equation involves concentrated sources; the boundary
conditions are mixed.
As is known, classical numerical methods lead us
to large errors that can exceed many times the exact solution
for small $\eps$; a similar problem occurs if we are to find
the normalized flux, i.e., the gradient multiplied
by $\eps$. New special schemes are constructed to converge
uniformly with respect to the parameter.
The errors in the discrete solution and in the computed fluxes
are independent of the parameter.
The new schemes can be applied to the analysis of heat
exchange in metal working by hot die-forming or
for plastic shear.
\end{abstract}
\section{Introduction}
Many modern technologies of material working are characterized by fast-running
processes. When modelling and analyzing heat and mass transfer, we
come to singularly perturbed equations in the dimensionless form.
For example, plastic shear in a material can be considered as
shifting two parts of the body under tangential stress. As a result,
on the slip surface heat is liberated. This process is described by such
parameters as the coefficient of temperature conductivity for steel
$D^H=2\cdot 10^{-5}\, m^2\cdot sec^{-1}$, the thickness of the shifting
parts $L=1\ m$; the duration of the process is $2\, sec$ when the shift
stage $\vartheta=1\,sec$. Then we come to the problem
with concentrated sources for $\eps=\eps_0=6.3\cdot 10^{-3}$
where $\eps^2=2D^HL^{-2}\vartheta$.
During the process, in a narrow neighbourhood
of the slip surface, temperature arises
significantly and becomes about some hundreds of Celsium degree. To
analyze the process, we are to find both the solution (e.g.,
the maximal temperature) and the heat fluxes, which determine the
structure of transformations in metal. Thus, singularly perturbed
problems are typical for fast-running processes.
For these problems the error in the discrete solution can be large
for small values of the parameter $\eps$ if we use
classical finite difference schemes (see, e.g., \cite{dool,mill,shi2}).
Therefore it is required to develop special methods for which the errors
do not depend on the parameter value.
For problems of plastic shear, with the use of the classical scheme
(see (\ref{eq6}), (\ref{eq10})) for $N=N_0=100$,
the growth of temperature is $835^{\ o}C$. For the special scheme
(see (\ref{eq6}), (\ref{eq11})) the growth is
$239^{\ o}C$, and the error is 7\%.
\section{Mathematical Formulation for the Problem}
On the set
\begin{equation} \label{eq1}
G=D\times (\,0,T\,], \quad D=\{\,x:\ d_00,\ c(x,t)\geq 0,\
p(x,t)\geq p_0>0,\; (x,t)\in\ovl{G}$.
We shall call the function
$$
P(x,t)=\eps\frac{\partial}{\partial x}\,u(x,t)
$$
the normalized diffusion flux. This function is
discontinuous on the set $S^*$.
It is convenient
to divide the set $\ovl{G}$ into two sets
$$
\ovl{G}=\ovl{G}^{\,1}\cup\ovl{G}^{\,2},\ \ G^1=D^1\times (\,0,T\,],
\ \ G^2=D^2\times (\,0,T\,] \enspace ,
$$
where $D^1=\{x:\ d_0