INVESTIGATION OF DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS BY USING COMPUTER ALGEBRA ALGORITHMS

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Resumo

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 construction. 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 difference schemes under consideration, the quadratic dependence of the error for large Reynolds numbers and the inversely proportional dependence for creeping currents are analyzed.

Sobre autores

Yu. BLINKOV

Chernyshevsky Saratov National Research State University; Peoples’ Friendship University of Russia; Joint Institute for Nuclear Research

Email: blinkovua@info.sgu.ru
Saratov, Russia; Moscow, Russia; Dubna, Moscow oblast, Russia

A. REBRINA

Gagarin State Technical University of Saratov

Autor responsável pela correspondência
Email: anrebrina@yandex.ru
Saratov, Russia

Bibliografia

  1. Самарский А.А. Теория разностных схем. 3-е изд., испр. – М. Наука. 1989. 616 с.
  2. Блинков Ю.А., Мозжилкин В.В. Генерация разностных схем для уравнения Бюргерса построением базисов Грёбнера // Программирование. 2006. № 2. С. 71–74.
  3. Gerdt V.P., Blinkov Yu.A., Mozzhilkin V.V. Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations // SIGMA. 2006. № 2(051). 26 p.
  4. Buchberger B. Gröbner bases: an Buchberger algorithmic method in polynomial ideal theory // Recent Trends in Multidimensional System Theory / Ed. by N.K. Bose. V. 6. Reidel, Dordrecht, 1985. P. 184–232.
  5. Шокин Ю.И., Яненко Н.Н. Метод дифференциального приближения. Применение к газовой динамике. Новосибирск: Наука: Сиб. отд-ние, 1985. 364 с.
  6. Блинков Ю.А., Гердт В.П., Маринов К.Б. Дискретизация квазилинейных эволюционных уравнений методами компьютерной алгебры // Программирование. 2017. № 2. С. 28–34.
  7. Zhang X., Gerdt V.P., Blinkov Y.A. Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow // Symmetry. 2019. № 11(269). 15 p.
  8. Блинков А.Ю., Малых М.Д., Севастьянов Л.А. О дифференциальных приближениях разностных схем // Изв. Сарат. ун-та. Нов. cер. Сер. Математика. Механика. Информатика. 2021. № С. 472–488.
  9. Gerdt V.P., Robertz D. Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification / In: S. Watt (ed.). Proceedings of ISSAC 2010 P. 53–59.
  10. Gerdt V.P. Consistency Analysis of Finite Difference Approximations to PDE Systems / Mathematical Modelling in Computational Physics 2011. LNCS. V. 7125. 2012. P. 28–42.
  11. Scala R.L. Gröbner bases and gradings for partial difference ideals // Math. Comput. 2015. № 84. P. 959–985.
  12. Gerdt V.P., Blinkov Yu.A. Involution and Difference Schemes for the Navier–Stokes Equations / In: Gerdt V.P., Mayr E.W., Vorozhtsov E.V. (eds.) CASC 2009. LNCS. 2009. V. 5743. P. 94–105.
  13. Amodio P., Blinkov Yu.A., Gerdt V.P., La Scala R. On Consistency of Finite Difference Approximations to the Navier-Stokes Equations / In: Gerdt V.P., Koepf W., Mayr E.W., Vorozhtsov E.V. (eds.) CASC 2013. LNCS. 2013. V. 8136. P. 46–60.
  14. Amodio P., Blinkov Yu.A., Gerdt V.P., Scala R.La. Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier–Stokes equations // Appl. Math. and Comput. 2017. № 314. P. 408–421.
  15. Blinkov Yu.A., Gerdt V.P., Lyakhov D.A., Michels D.L. A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations / In: Gerdt V.P., Koepf W., Mayr E.W., Vorozhtsov E.V. (eds.) CASC 2018. LNCS. 2018. V. 11077. P. 67–81.
  16. Michels D.L., Gerdt V.P., Blinkov Y.A., Lyakhov D.A. On the Consistency Analysis of Finite Difference Approximations // Journal of Mathematical Sciences (United States). 2019. № 5. P. 665–677.
  17. Gerdt V.P., Robertz D., Blinkov Yu.A. Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations / eprint arXiv:2009.01731. 2019. 48 p.
  18. Cartan E. Sur certaines expressions différentielles et le problème de Pfaff // Annales scientifiques de l’École Normale Supérieure Sér. 3. 1899. № 16. P. 239–332.
  19. Kähler E. Einführung in die Theorie der Systeme von Differentialgleichungen. Hamburger mathematische Einzelschriften 16. Leipzig: B.G. Teubner. 1934. Vol IV. 80 p.
  20. Riquier C. Les Systèmes d’Equations aux Dérivées Partielles. Mémorial Sci. Math. XXXII, Gauthier-Villars, Paris. 1910.
  21. Harlow F.H., Welch J.E. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free // Phys. Fluids. 1965. № 8. P. 2182–2189.
  22. Buchberger B. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal: Ph.D. thesis / Universiät Innsbruck. 1965.
  23. Taylor G.I. On the decay of vortices in a viscous fluid // Philosophical Magazine. 1923. V. 46. P. 671–674.
  24. Kovasznay L.I.G. Laminar flow behind a two-dimensional grid // Mathematical Proceedings of the Cambridge Philosophical Society. 1948. V. 44. № 1. P. 58–62.

Declaração de direitos autorais © Ю.А. Блинков, А.Ю. Ребрина, 2023

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