Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 184–196. ISSN 2079-6641

Contents of this issue

Read Russian Version US Flag

MSC 76M20

Research Article

Locally one-dimensional schemes for an equation describing coagulation processes in convective clouds with “memory”

M. H. Shhanukov-Lafishev¹, M. M. Lafisheva², I. D. Taisaev³

¹Institute of Applied Mathematics and Automation of KBSC RAS, 360000, Kabardino-Balkar Republic, Nalchik, Shortanova str., 89-A, Russia
²Kabardino-Balkar State University, 360004, Kabardino-Balkar Republic, Nalchik, Cherhyshevskaya str., 179, Russia
³Kabardin-Balkar Scientific Center of KBSC RAS, 360002, Kabardino-Balkar Republic, Nalchik, Balkarova str., 2, Russia


The paper considers a locally one-dimensional difference scheme for a general parabolic equation in a p-dimensional parallelepiped. To describe coagulation processes in media with “memory”, non-local sources of a special type are included in the equation. An a priori estimate is obtained for solving the corresponding difference scheme, which implies its convergence.

Key words: boundary value problem, locally one-dimensional difference scheme, stability and convergence of the difference scheme, approximation error.

DOI: 10.26117/2079-6641-2022-39-2-184–196

Original article submitted: 29.04.2022

Revision submitted: 28.08.2022

For citation. Shhanukov–Lafishev M. H., Lafisheva M. M., Taisaev I. D. Locally onedimensional schemes for an equation describing coagulation processes in convective clouds with “memory”. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 184–196. DOI: 10.26117/2079-6641-2022-39-2-184–196

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

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

© Shhanukov-Lafishev M. H., Lafisheva M. M., Taisaev I. D., 2022


  1. Ashabokov B. A., Taisaev I. D., Shkhanukov-Lafishev M. Kh. A local onedimensional scheme for parabolic equation of general form, describing microphysical processes in convective clouds, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 23: 3, 158-167,(In Russian)
  2. Kogan E. L., Mazin I.P., Sergeev B. N., Khvorostyanov V. I. [Numerical modeling of clouds], Moscow, Gidrometeoizdat, 1984, p. 178.(In Russian)
  3. Ashabokov B. A., Shapovalov A. V. [Convective Clouds: Numerical Models and Simulation Results in Natural Conditions and Under Active Influence], Nalchik, Publishing House of KBSC RAS, 2008. p. 254.(In Russian)
  4. Berry E. X. Gloud Droplet by Collection, J. Atoms Sci., 1967, vol. 24, pp. 688-701.
  5. Berry E. X., Reinhardt R. L. An analysis of Gloud Drop Grouth by Collection, J. Atoms Sci., 1974, vol. 24, pp. 1825–1837.
  6. Samarskii A. A. The Theory of Difference Schemes (1st ed.), New York, CRC Press, 2001, p. 786. DOI: 10.1201/9780203908518
  7. Andreev V. B. Convergence of difference schemes approximating the second and third boundary value problems for elliptic equations, Journal of Computational Mathematics and Mathematical Physics, 1968. vol. 8, no. 6, pp. 1218–1231 (In Russian)
  8. Samarskiy А. A., Gulin A. V. Ustoychivost’ raznostnykh skhem [Stability of difference schemes], Moscow, Nauka, 1973, p. 480 (In Russian)

Shhanukov–Lafishev Mukhamed Habalovich – D. Sci. (Phys. & Math.), Professor, Institute Of Applied Mathematics and Automation of KBSC RAS, Nalchik, Russia,, ORCID 0000- 0002-7242-975X.

Lafisheva Madina Mukhammedovna – Ph. D. (Phys. & Math.), Associate Professor of the Department of Applied Mathematics and Informatics, Institute of Artificial Intelligence and Digital Technologies, Kabardino-Balkarian State University named after Kh. M. Berbekov, Nalchik, Russia,, ORCID 0000-0002-7230-6629.

Taysaev Idar Djabrailovich – Graduate student of Kabardin–Balkar Scientific Center of Russian Academy of Sciences, Nalchik, Russia,, ORCID 0000-0002-0191-8353.