Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 104 – 123. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2026-54-1-104-123
Research Article
Full text in Russian
MSC 65M06, 65M12, 74F10, 76D05

Contents of this issue

Read Russian Version

Symbolic Investigation and Numerical Experiments on a Finite-Difference Scheme for Wave Processes in Coaxial Cylindrical Shells Filled with an Incompressible Viscous Fluid

A. V. Mesyanzhin∗

JSC “Design Bureau of Industrial Automatics 239 B. Sadovaya St, Saratov, 410005, Russian Federation

Abstract. Wave processes in a system of two coaxial infinitely long cylindrical elastic shells, the space between which is filled with a viscous incompressible fluid, are studied. Based on the coupled hydroelasticity problem and the perturbation method, a mathematical model in the form of a system of generalized Korteweg–de Vries equations is obtained. For its numerical solution, an implicit conservative difference scheme of the Crank–Nicolson type is constructed using Grobner bases. The quality of the scheme is analyzed by the method of the first differential approximation, which makes it possible to estimate the local approximation error. To analyze the global error, an error accumulation function is proposed and applied. The theoretical analysis is supplemented by numerical experiments that investigate the transfer of a wave disturbance from the outer shell to the inner one at various fluid parameters. It is shown that the error of the scheme is localized mainly on the wave fronts, depends on the wavenumber and nonlinearity parameters, and remains controllable during the calculations. The results confirm the correctness, stability, and efficiency of the proposed difference scheme for modeling wave processes in the considered complex mechanical systems.

Key words: Gr¨obner bases, difference schemes, first differential approximation, Korteweg–de Vries equation, nonlinear shells, viscous fluid

Received: 30.12.2025; Revised: 04.02.2026; Accepted: 07.02.2026; First online: 29.03.2026

For citation. Mesyanzhin A. V. Symbolic Investigation and Numerical Experiments on a Finite-Difference Scheme for Wave Processes in Coaxial Cylindrical Shells Filled with an Incompressible Viscous Fluid. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 104-123. EDN: RWPRSV. https://doi.org/10.26117/2079-6641-2026-54-1-104-123.

Funding. No funding was received.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author contributed to this article and is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.

^{\ast}Correspondence: E-mail: a.v.mesyanzhin@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Mesyanzhin A. V., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

References

  1. Gorshkov A. G., Medvedskiy A. L., Rabinskiy L. N., Tarlakovskiy D. V.Waves in continuous media, Moscow: Fizmatlit, 2004, 472 p.
  2. Kudryashov N. A. Methods of nonlinear mathematical physics, Dolgoprudny: Intellect, 2010, 368 p.
  3. Nariboli G. A., Sedov A. Burgers’s-Korteweg-De Vries equation for viscoelastic rods and plates, J. Math. Anal. Appl., 1970, vol. 32, pp. 661–677. DOI: 10.1016/0022-247X(70)90290-8
  4. Erofeev V. I., Kazhaev V. V. Inelastic interaction and splitting of deformation solitons propagating in the rod, Computational Continuum Mechanics, 2017, vol. 10, no. 2, pp. 127–136. DOI: 10.7242/1999-6691/2017.10.2.11
  5. Bochkarev A. V., Zemlyanukhin A. I., Mogilevich L. I. Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium, Acoust. Phys., 2017, vol. 63, pp. 148–153. DOI: 10.1134/S1063771017020026
  6. Zemlyanukhin A. I., Bochkarev A. V., Andrianov I. V., Erofeev V. I. The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells, J. Sound Vib., 2021, vol. 491. 115752 p. DOI: 10.1016/j.jsv.2020.115752
  7. Gromeka I. S. On the velocity of wave-like motion of fluids in elastic tubes, CollectedWorks, Moscow: Izd-vo AN USSR, 1952, pp. 172–183.
  8. Zhukovskiy N. E. On hydraulic shock in water pipes, Moscow: Gostekhizdat, 1949, 103 p.
  9. Womersley J. R. Oscillatory motion of a viscous liquid in a thin-walled elastic tube. I. The linear approximation for long waves, Phil. Mag., 1955, vol. 46, pp. 199–221. DOI: 10.1080/14786440208520564
  10. Pa¨ıdoussis M.P. Fluid-structure interactions. Slender structures and axial flow. Vol. 2, London: Elsevier Academic Press, 2016, 942 p. DOI: 10.1016/C2011-0-08058-4
  11. Blinkova A.Yu., Blinkov Yu. A., Mogilevich L. I. Non-linear waves in coaxial cylinder shells containing viscous liquid inside with consideration for energy dispersion, Computational Continuum Mechanics, 2013, vol. 6, no. 3. 336–345 p. DOI: 10.7242/1999-6691/2013.6.3.38
  12. Blinkov Y. A., Mesyanzhin A. V., Mogilevich L. I. Nonlinear wave propagation in coaxial shells filled with viscous liquid, Computational Continuum Mechanics, 2017, vol. 10, no. 2, pp. 172–186. DOI: 10.7242/1999-6691/2017.10.2.15
  13. Bochkarev S. A., Matveenko V.P. Stability of coaxial cylindrical shells containing a rotating fluid, Computational Continuum Mechanics, 2013, vol. 6, no. 1, pp. 94–102. DOI: 10.7242/1999-6691/2013.6.1.12
  14. Volmir A. S. Nonlinear dynamics of plates and shells, Moscow: Nauka, 1972, 432 p.(In
    Russian).
  15. Volmir A. S. Shells in fluid and gas flow: hydroelasticity problems, Moscow: Nauka, 1979, 320 p.(In Russian).
  16. Blinkov Yu. A., Kondratova Yu. N., Mesyanzhin A. V., Mogilevich L. I. Mathematical modeling of nonlinear waves in coaxial shells filled with viscous fluid, Izv. Sarat. Univ. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 2016, vol. 16, no. 3, pp. 331–336. DOI: 10.18500/1816-9791-2016-16-3-331-336(In Russian).
  17. Slyunyaev A. V., Pelinovski E. N. Dynamics of large-amplitude solitons, J. Exp. Theor. Phys., 1999, vol. 89, no. 1, pp. 173–181. DOI: 10.1134/1.558966
  18. Crank J., Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proceedings of the Cambridge Philosophical Society, 1947, vol. 43, no. 1, pp. 50–67. DOI: 10.1017/S0305004100023197
  19. Gerdt V.P., Blinkov Yu. A., Mozzhilkin V. V. Gr¨obner Bases and Generation of Difference Schemes for Partial Differential Equations, Symmetry, Integrability and Geometry: Methods and Applications, 2006, vol. 2. 26 p. DOI: 10.3842/SIGMA.2006.051
  20. Blinkov Y. A., Gerdt V.P., Marinov K. B. Discretization of quasilinear evolution equations by computer algebra methods, Programming and Computer Software, 2017, vol. 43, no. 2, pp. 84–89. DOI: 10.1134/S0361768817020049
  21. Yanenko N. N., Shokin Yu. I. First differential approximation of difference schemes for hyperbolic systems of equations, Siberian Mathematical Journal, 1969, vol. 10, pp. 868–880. DOI: 10.1007/BF00971662
  22. Shokin Yu. I., Yanenko N. N. The Method of Differential Approximation. Application to Gas Dynamics, Nauka, 1985, 368 p.
  23. Gerdt V.P., Robertz D. Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC), 2010, pp. 53–59.
  24. Gerdt V.P. Consistency Analysis of Finite Difference Approximations to PDE Systems, Mathematical Modelling in Computational Physics 2011, 2012, vol. 7125, pp. 28–42.
  25. La Scala R. Gr¨obner bases and gradings for partial difference ideals, Mathematics of Computation, 2015, vol. 84, no. 293, pp. 959–985.
  26. Blinkova A.Yu., Malykh M. D., Sevast’yanov L. A. On differential approximations of difference schemes, Izv. Sarat. Univ. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 2021, vol. 21, no. 4, pp. 472–488. DOI: 10.18500/1816-9791-2021-21-4-472-488 (In Russian).
  27. Blinkov Y. A., Rebrina A. Y. Investigation of Difference Schemes for Two-Dimensional Navier–Stokes Equations by Using Computer Algebra Algorithms, Programming and Computer Software, 2023, vol. 49, no. 1, pp. 26–31. DOI: 10.1134/S0361768823010024
  28. Blinkov Yu. A. Qualitative investigation of Crank-Nicolson-type schemes using the firstorder differential approximation on the example of the Korteweg-de Vries equation, Mathematical Modeling, Computational and Experimental Studies in Natural Sciences, 2023, no. 4, 18 p. DOI: 10.24412/2541-9269-2023-4-12-28
  29. Blinkov Yu. A. Symbolic and numerical investigation of finite difference schemes using the first differential approximation: the case of the Korteweg-de Vries equation, Mathematical Modeling, Computational and Experimental Studies in Natural Sciences, 2024, no. 4, 13 p. DOI: 10.24412/2541-9269-2024-4-20-32
  30. Blinkov Y. A., Gerdt V.P. Specialized computer algebra system GINV, Programming and Computer Software, 2008, vol. 34, no. 2, pp. 112–123. DOI: 10.1134/S0361768808020096

Information about the author

Mesyanzhin Artem Vyachaslavovich – Head of the Laboratory Sector, JSC “Design Bureau of Industrial Automation Saratov, Russian Federation, ORCID 0000-0002-7984-2168.

Read articleDownload article