The present paper deals with monotone difference schemes of second order of accuracy for the approximation of differential boundary value problems. Both boundary conditions of the second and third kind are considered without using the differential equation at the domain of the boundary. The idea is based on the assumption of the existence and uniqueness of a smooth solution in some sufficiently small neighborhoods of the definition domain and the use of half-integer nodes of the grid. Then, second-order schemes are constructed.
Moreover, if the equation has also a meaning at the boundary nodes, then fourth-order monotone schemes can be derived. A special attention is given to the Neumann problem. Finally, the construction of second-order difference schemes for a two-dimensional parabolic equation are obtained. No numerical results are given.