On compact approximations of divergent differential equations

The method for construction of compact difference schemes approximating the divergent differential equations is proposed. The schemes have an arbitrary order of approximation on a stencil of a common type. It is shown that construction of such schemes for partial differential equations is based on spatial compact schemes approximating ordinary differential equations depending on several independent functions. Necessary and sufficient conditions on factors of these schemes, at which they have a high order of approximation, are obtained. Examples of restoration with these schemes of compact difference schemes for partial differential equations are given. It is shown that such compact difference schemes have the same order as classical approximation on smooth solutions and weak approximations on discontinuous solutions.