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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.

This paper explores the numerical intergator of ODE based on combination of Appelroth's quadratization of dynamical systems with polynomial right-hand sides and Kahan's discretization method. Utilizing Appelroth's technique, we reduce any system of ordinary differential equations with a polynomial right-hand side to a quadratic form, enabling the application of Kahan's method. In this way, we get a difference scheme defining the one-to-one correspondence between the initial and final positions of the system (Cremona map). It provides important information about the Kahan method for differential equations with a quadratic right-hand side, because we obtain dynamical systems with a quadratic right-hand side that have movable branch points. We analyze algebraic properties of solutions obtained through this approach, showing that (1) the Kahan scheme describes the branch points as poles, significantly deviating from the behavior of the exact solution of the problem near these points, and (2) it disrupts algebraic invariant variety, in particular integral relations describing the relationship between old and Appelroth's variables. This study advances numerical methods, emphasizing the possibility of designing difference schemes whose algebraic properties differ significantly from those of the initial dynamical system. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson's method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method. [ABSTRACT FROM AUTHOR]