Рresented by Academician of the NAS of Ukraine P.I. Andon Mathematical problems associated with the numerical solution of equations for the predictive models of regional atmospheric circulation are considered. A methodology is considered for effective regional solutions of boundary-value problems with a prehistory on the basis of the "one-way interaction" approach. Within this approach, a method is proposed for filling the data given on the macroscale grid nodes in the mesoscale network based on the spline interpolation and a precise (the fourth-order of accuracy) numerical method for the approximation of the first-and second-order derivatives in differential equations. Thereby, solving problems for ordinary differential equations can be carried out effectively by the interpolation.