On differential approximations of difference schemes

The concept of the first differential approximation was introduced in the 1950s for the analysis of difference schemes by A. I. Zhukov and then was used to study the quality of difference schemes approximating equations in partial derivatives. In the present work, the first differential approximation is considered as a universal construction that allows to use computer algebra methods for investigation difference schemes, bypassing the direct use of the methods of difference algebra. The first section discusses the differential approximation for difference schemes approximating ordinary differential equations. The relationship between differential approximation, singular perturbation of the original system and the concept of the first differential approximation is discussed. For this simple case, the estimation for the difference between exact and approximate solutions is given and justified, the method is compared with Richardson - Kalitkin method. The second section discusses differential approximations for difference schemes approximating partial differential equations. The concept of the first differential approximation is described in the language of power geometry. As it has been shown, when approximating a consistent system of partial differential equations, consistent difference systems of equations are not always obtained. As a method of checking the consistency of a difference equations system, it is proposed to check the consistency of the first differential approximation for the difference system. From this point of view, the concept of strong consistency (s-consistency) of a system of difference equations is discussed. A few examples of systems that are not strongly consistent are given. To analyse the consistency of the first differential approximation, software developed for the investigation of partial differential equations is used. The problem of calculation of the first differential approximation in computer algebra, Sage and SymPy systems is considered.