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{\bf EXPLICIT-IMPLICIT DIFFERENCE SCHEMES FOR CONVECTION-DIFFUSION PROBLEMS}

{\bf A.A.~Samarskii and P.N.~Vabishchevich}

{\it Institute for Mathematical Modelling,
Russian Academy of Sciences,

4 Miusskaya Square,
Moscow 125047,
Russian Federation
}
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Convection-diffusion problems are basic ones in modelling problems
of hydrodynamics and heat transfer. Primary peculiarities of
these problems are connected with the nonseldadjoint property of
an elliptic operator of the problem, i.e. domination of convective
transport. In numerical solution the emphasis is on issues of
approximation of convective terms. There are in common use upwind
schemes, hybrid schemes or high order directed schemes.

Nowadays properties of employed difference schemes are investigated
basically via numerical experiments on some test problems. A theoretical
study is traditionally conducted using the maximum principle.
Methods with the Fourier transformation or the Neumann approach are
employed, too. Unfortunately, only the simplest one-dimensional
convection-diffusion problems with constant coefficients can be
analyzed using the above techniques.

An alternative approach developed by A.A.~Samarskii and his
pupils for a wide class of difference schemes is based on the
general theory of stability of operator-difference schemes
which are considered in the corresponding finite-dimensional
Hilbert spaces. Using this approach there are derived
unimproved (coincide necessary and sufficient) conditions
of stability for a wide class of two- and three-level schemes.
Constructibility of this theory results from the fact that
stability conditions are formulated in the form of easy checking
operator inequalities.

In the preset work there is conducted a study on some class of
difference schemes for convection-diffusion problems using the general
theory of stability. Explicit-implicit schemes where convective transport
is taken from the previous time-level are considered. The Dirichlet
problem for the time-dependent convection-diffusion problem is studied
in a rectangular. Convective terms are considered in the divergent
or nondivergent forms and are approximated with the second order
using the corresponding central difference derivatives.
Apriori estimates consistent with estimates for the solution of
the differential problem are derived for the difference solution.
A class of unconditionally stable explicit-implicit schemes
is constructed.

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