  This work deals with the problem of choosing a time step for the numerical
solution of boundary value problems for parabolic equations. The problem
solution is derived using the fully implicit scheme, whereas a time step is
selected via explicit calculations. The selection strategy consists of the
following steps. First, using the explicit scheme, we calculate the solution at
a new time level. Next, we employ this solution in order to obtain the solution
at the previous time level (the implicit scheme, explicit calculations). This
solution should be close to the solution of our problem at this time level with
a prescribed accuracy. Such an algorithm leads to explicit formulas for the
calculation of the time step and takes into account both the dynamics of the
problem solution and changes in coefficients of the equation and in its
right-hand side. The same formulas for the evaluation of the time step we get
using a comparison of two approximate solutions, which are obtained using the
explicit scheme with the primary time step and the step that is reduced by
half. Numerical results are presented for a model parabolic boundary value
problem, which demonstrate the robustness of the developed algorithm for the
time step selection.
