On compact approximations of divergence differential equations

Abstract

A method for the construction of compact difference schemes approximating divergence differential equations is proposed. The schemes have an arbitrarily prescribed order of approximation on general stencils. It is shown that the construction of such schemes for partial differential equations is based on special compact schemes approximating ordinary differential equations in several independent functions. Necessary and sufficient conditions on the coefficients of these schemes with high order of approximation are obtained. Examples of reconstruction of compact difference schemes for partial differential equations with these schemes are given. It is shown that such compact difference schemes have the same order of accuracy both for classical approximations on smooth solutions and weak approximations on discontinuous solutions.