Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Blinkov Y. A., Malykh M. D., Sevastianov L. A. On differential approximations of difference schemes. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 4, pp. 472-488. DOI: 10.18500/1816-9791-2021-21-4-472-488, EDN: BBTOTY

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On differential approximations of difference schemes

Blinkov Yuri Anatolievich, Saratov State University
Malykh Mikhail D., Peoples’ Friendship University of Russia (RUDN University)
Sevastianov Leonid A., Peoples’ Friendship University of Russia (RUDN University)

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 work was supported by the Russian Science Foundation (project No. 20-11-20257).
  1. Godunov S. K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik, 1959, vol. 47, no. 3, pp. 271–306 (in Russian).
  2. Yanenko N. N., Shokin Yu. I. The first differential approximation of difference schemes for hyperbolic systems of equations. Siberian Mathematical Journal, 1969, vol. 10, no. 5, pp. 868–880. https://doi.org/10.1007/BF00971662
  3. Shokin Yu. I. Metod differentsial’nogo priblizheniya [Differential Approximation Method]. Novosibirsk, Nauka, AN SSSR, 1979. 222 p. (in Russian).
  4. Shokin Yu. I., Yanenko N. N. Metod differencial’nogo priblizhenija. Primenenie k gazovoj dinamike [Differential Approximation Method. Application to Gas Dynamics]. Novosibirsk, Nauka, AN SSSR, 1985. 364 p. (in Russian).
  5. Levin A. Difference Algebra. Springer, 2008. 521 p. https://doi.org/10.1007/978-1-4020-6947-5
  6. van der Put M., Singer M. F. Galois Theory of Difference Equations. Springer, 1997. 188 p. https://doi.org/10.1007/BFb0096118
  7. Hendriks P. A. Algebraic aspects of linear differential and difference equations. University of Groningen, 1996. PhD thesis. 106 p.
  8. Weierstrass K. Uber der Theorie der analytischen Facultaten. Mathematische werke. Berlin : Mayer & Muller, 1894, vol. 1, pp. 153–221.
  9. Grammaticos B., Nijhoff F. W., Ramani A. Discrete Painleve equations. In: The Painleve Property, One Century Later. Berlin, Heidelberg, Springer, 1999, pp. 413–516.
  10. Vasil’eva A. B., Butuzov V. F. Asimptoticheskie metody v teopii singuljapnyh vozmushсhenij [Asymptotic Methods in the Theory of Singular Perturbations]. Moscow, Vysshaja shkola, 1990. 208 p. (in Russian).
  11. Kalitkin N. N., Al’shin A. B., Al’shina E. A., Rogov B. V. Vychislenija na kvaziravnomernykh setkakh [Calculations on Quasi-uniform Grids]. Moscow, Nauka, Fizmatlit, 2005. 204 p. (in Russian).
  12. Bruno A. D. Reshenie algebraicheskogo uravnenija algoritmami stepennoj geometrii [Solving an Algebraic Equation by Algorithms of Power Geometry]. Moscow, Nauka, 1998. 288 p. (in Russian).
  13. Bruno A. D. Power Geometry in Algebraic and Differential Equations. Preprinty IPM No. 34. Moscow, IPM imeni Keldysha, 2017, pp. 1–18 (in Russian).
  14. Zhang Xiaojing, Gerdt V. P., Blinkov Yu. A. Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow. Symmetry, 2019, vol. 11, no. 2. Available at: https://www.mdpi.com/2073-8994/11/2/269 (accessed 10 May 2021).
  15. Robertz D. Formal Algorithmic Elimination for PDEs. Springer, 2014. 284 p.
  16. Hans Johnston, Jian-Guo Liu. Finite difference schemes for incompressible flow based on local pressure boundary. Journal of Computational Physics, 2002, no. 180, pp. 120–154.