The monograph identifies a new class of difference schemes with operator multipliers, to which the schemes with variable weight multipliers belong. The theory of stability of such two- and three-layer difference schemes with non-self-adjoint operators develops. Based on the obtained general results, a study was conducted on the stability and convergence of various initial-edge problems for partial-derivative equations. Adaptive difference schemes are built with the condensation of grids in time and space. A theoretical analysis of regional-additive difference schemes (area decomposition schemes) focused on the construction of effective computational algorithms for parallel computing systems is carried out. The book is designed for specialists in numerical methods of solving problems of mathematical physics, the material is available to senior students of universities .