The present paper deals with a generalization of the alternating-direction implicit
(ADI) method for a two dimensional Poisson equation in a rectangle domain with a
weighted integral boundary condition in one coordinate direction. We consider the alternating
direction method for a system of difference equations that approximates Poisson equation
with weighed integral boundary conditions with the fourth-order accuracy. Sufficient conditions
of stability for ADI method are investigated numerically. An analysis of results of
computational experiments is presented.