  To solve numerically boundary value problems for parabolic equations with
mixed derivatives, the construction of difference schemes with prescribed
quality faces essential difficulties. In parabolic problems, some possibilities
are associated with the transition to a new formulation of the problem, where
the fluxes (derivatives with respect to a spatial direction) are treated as
unknown quantities. In this case, the original problem is rewritten in the form
of a boundary value problem for the system of equations in the fluxes. This
work deals with studying schemes with weights for parabolic equations written
in the flux coordinates. Unconditionally stable flux locally one-dimensional
schemes of the first and second order of approximation in time are constructed
for parabolic equations without mixed derivatives. A peculiarity of the system
of equations written in flux variables for equations with mixed derivatives is
that there do exist coupled terms with time derivatives.
