26 ISSN 0361-7688, Programming and Computer Software, 2023, Vol. 49, No. 1, pp. 26–31. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Programmirovanie, 2023, Vol. 49, No. 1. Investigation of Difference Schemes for Two-Dimensional Navier–Stokes Equations by Using Computer Algebra Algorithms Yu. A. Blinkov a,b,c, * (ORCID: 0000-0001-7340-0919) and A. Yu. Rebrina d, ** a Chernyshevsky Saratov National Research State University, ul. Astrakhanskaya 83, Saratov, 410012 Russia b Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198 Russia c Joint Institute for Nuclear Research, ul. Zholio-Kyuri 6, Dubna, Moscow oblast, 141980 Russia d Gagarin State Technical University of Saratov, ul. Politekhnicheskaya 77, Saratov, 410054 Russia *e-mail: blinkovua@info.sgu.ru **e-mail: anrebrina@yandex.ru Received July 1, 2022; revised August 23, 2022; accepted September 7, 2022 Abstract—A class of consistent difference schemes for incompressible Navier–Stokes equations in physical variables and their differential approximations are considered using an algorithm for Gröbner basis construc- tion. Results of investigating the first differential approximations of these schemes, which are obtained by using the authors' programs implemented in the SymPy computer algebra system, are presented. For the dif- ference schemes under consideration, the quadratic dependence of the error for large Reynolds numbers and the inversely proportional dependence for creeping currents are analyzed. DOI: 10.1134/S0361768823010024 1. INTRODUCTION Exact solutions to systems of partial differential equations can presently be found only in very rare cases, which is why, in mechanics and physics, these systems have to be solved numerically, i.e., by replac- ing them with their discrete counterparts. The most popular discretization and numerical methods are finite element, finite volume, and finite difference methods. The last method was historically the first [1]; it is based on the replacement of differential equations with difference equations defined on a selected grid. To construct a numerical solution, a finite-difference approximation of partial differential equations is sup- plemented with an appropriate discretization of initial and/or boundary conditions. In this case, the similar- ity of numerical and exact solutions of a system of par- tial differential equations depends heavily on the qual- ity of a difference scheme. In [2, 3], an algorithm for constructing conserva- tive difference schemes was developed and imple- mented in Maple. The desired system is represented as relational conditions for difference functions, their difference derivatives are represented as integral rela- tions, and the corresponding ranking is selected by computing the difference Gröbner basis, which adapts the well-known algorithm for investigation of polyno- mial systems to the difference case [4]. As a result of applying the algorithm for constructing the Gröbner basis of a difference ideal for differential equations with the derivatives of order higher than one, a differ- ence scheme is constructed in the form of a relation only on the functions themselves. In this case, the dif- ference scheme obtained is checked for consistency and its dimension (solution space) is determined. One of the algorithmic approaches for investigation of difference schemes is the construction of the first differential approximation [5]. Using this approach and Gröbner bases, differential approximations were constructed for equations of evolutionary type [6] and for systems of equations [7]. In [8], some problems of computing the first differential approximation in com- puter algebra systems were considered. If a difference scheme is not consistent or has a smaller solution space as compared to the original system of differential equations, then, when constructing the first differen- tial approximation, extra equations occur and compu- tations can be terminated. This check is much more efficient (in terms of both time and memory) than the consistency check in the difference case, and it can easily be implemented using standard tools of com- puter algebra systems.