This monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Pade approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp - that is, two-sided - estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. The book should be of value to professional mathematicians, as well as advanced students in the fields of functional analysis and partial differential equations.