  Various classes of stable finite difference schemes can be constructed to
obtain a numerical solution. It is important to select among all stable schemes
such a scheme that is optimal in terms of certain additional criteria. In this
study, we use a simple boundary value problem for a one-dimensional parabolic
equation to discuss the selection of an approximation with respect to time. We
consider the pure diffusion equation, the pure convective transport equation
and combined convection-diffusion phenomena. Requirements for the
unconditionally stable finite difference schemes are formulated that are
related to retaining the main features of the differential problem. The concept
of SM stable finite difference scheme is introduced. The starting point are
difference schemes constructed on the basis of the various Pad$\acute{e}$
approximations.
