  The problem of increasing the accuracy of an approximate solution is
considered for boundary value problems for parabolic equations. For ordinary
differential equations (ODEs), nonstandard finite difference schemes are in
common use for this problem. They are based on a modification of standard
discretizations of time derivatives and, in some cases, allow to obtain the
exact solution of problems. For multidimensional problems, we can consider the
problem of increasing the accuracy only for the most important components of
the approximate solution. In the present work, new unconditionally stable
schemes for parabolic problems are constructed, which are exact for the
fundamental mode. Such two-level schemes are designed via a modification of
standard schemes with weights using Pad\'{e} approximations. Numerical results
obtained for a model problem demonstrate advantages of the proposed fundamental
mode exact schemes.
