  The method of monotonization of difference schemes is being considered in the
paper. The method was earlier proposed by the author for stationary problems.
It is investigated in the paper more profoundly. The idea of the method is to
build the monotonizing operators into the schemes so that the balance relations
from point to point are not violated. Different monotonizing operators can be
used to be installed in the schemes. Propositions concerning approximation and
stability of the monotonized schemes are formulated and proved. Also a
proposition significant for practical use of the schemes is formulated and
proved. The idea is to use the monotonized schemes in the cases when the
proposition conditions are fulfilled. The proposition is based on closeness of
solutions of the initial and auxiliary schemes. Constructions for solving of
time dependent problems are also written in the paper. One dimensional example
and three-dimensional hydrodynamic example are considered. The method allows to
considerably decrease value of calculations in many cases.
