Study of an implicit iterative scheme for a numerical solution of stationary heat convection equations based on the idea of "weak compressibility"

Perizat Beisebay; Dinara Omariyeva; Gulmira Kenzhebekova; Dauren Matin; Gabit Mukhamediyev; Perizat Beisebay; Dinara Omariyeva; Gulmira Kenzhebekova; Dauren Matin; Gabit Mukhamediyev

doi:10.3934/math.2026456

AIMS Mathematics

2026, Volume 11, Issue 4: 11074-11098. doi: 10.3934/math.2026456

Previous Article Next Article

Research article

Study of an implicit iterative scheme for a numerical solution of stationary heat convection equations based on the idea of "weak compressibility"

1.
Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan
2.
International School of Engineering, D. Serikbayev East Kazakhstan Technical University, Ust-Kamenogorsk 070000, Kazakhstan
3.
Higher School of IT and Natural Sciences, S. Amanzholov East Kazakhstan University, Ust-Kamenogorsk 070000, Kazakhstan

Received: 25 January 2026 Revised: 01 April 2026 Accepted: 08 April 2026 Published: 21 April 2026
MSC : 65N30, 65N12

This paper investigates a class of fully implicit iterative schemes for the numerical solution of stationary thermal convection equations formulated in primitive variables (velocity-pressure-temperature). The proposed approach is based on the concept of weak compressibility, which relaxes the incompressibility constraint within a pseudotime-iterative framework. From an algorithmic perspective, the resulting scheme can be interpreted as a fully implicit realization within the broader class of projection and pressure-correction methods widely used in computational fluid dynamics. A rigorous theoretical analysis of the discrete problem is presented. A priori estimates guaranteeing the stability of the numerical solution are derived, and a uniqueness condition for the discrete convection problem is established. For the linear Stokes case, the convergence of the iterative algorithm is proven, and it is shown that the iteration process converges with a geometric rate whose constants are independent of the spatial grid step. Numerical experiments are performed for the classical natural convection benchmark problem in a square cavity. The results demonstrate grid convergence of the numerical solution, confirm the geometric convergence rate predicted by the theory, and illustrate the influence of the Rayleigh number and pseudotime-parameters on the behavior of the algorithm. The obtained results provide a rigorous mathematical foundation for implicit iterative algorithms closely related to projection-type methods and demonstrate their practical applicability to numerical simulation of convective flows.
- natural convection,
- primitive variables,
- pressure-velocity coupling,
- artificial compressibility,
- finite difference method,
- Navier-Stokes equations,
- implicit iterative schemes
Citation: Perizat Beisebay, Dinara Omariyeva, Gulmira Kenzhebekova, Dauren Matin, Gabit Mukhamediyev. Study of an implicit iterative scheme for a numerical solution of stationary heat convection equations based on the idea of "weak compressibility"[J]. AIMS Mathematics, 2026, 11(4): 11074-11098. doi: 10.3934/math.2026456

Related Papers:

Abstract

This paper investigates a class of fully implicit iterative schemes for the numerical solution of stationary thermal convection equations formulated in primitive variables (velocity-pressure-temperature). The proposed approach is based on the concept of weak compressibility, which relaxes the incompressibility constraint within a pseudotime-iterative framework. From an algorithmic perspective, the resulting scheme can be interpreted as a fully implicit realization within the broader class of projection and pressure-correction methods widely used in computational fluid dynamics. A rigorous theoretical analysis of the discrete problem is presented. A priori estimates guaranteeing the stability of the numerical solution are derived, and a uniqueness condition for the discrete convection problem is established. For the linear Stokes case, the convergence of the iterative algorithm is proven, and it is shown that the iteration process converges with a geometric rate whose constants are independent of the spatial grid step. Numerical experiments are performed for the classical natural convection benchmark problem in a square cavity. The results demonstrate grid convergence of the numerical solution, confirm the geometric convergence rate predicted by the theory, and illustrate the influence of the Rayleigh number and pseudotime-parameters on the behavior of the algorithm. The obtained results provide a rigorous mathematical foundation for implicit iterative algorithms closely related to projection-type methods and demonstrate their practical applicability to numerical simulation of convective flows.

References

[1]	E. Tarunin, Computational experiment in free convection problems (Russian), Irkutsk: Irkutsk University Publishing House, 1990.
[2]	P. Beisebay, N. Danaev, On one difference scheme for thermal convection equations (Russian), Bulletin of Abai KazNPU. Series: Physical and Mathematical Sciences, 1 (2007), 56–61.
[3]	P. Beisebay, N. Danaev, On the numerical solution of free convection with lateral heating (Russian), Bulletin of KazNU. Series: Mathematics, Mechanics, Computer Science, 1 (2007), 71–80.
[4]	A. Samarskii, The theory of difference schemes, New York: Marcel Dekker, Inc., 1977.
[5]	A. Samarskii, P. Vabishchevich, Computational heat transfer, New York: John Wiley & Sons, 1995.
[6]	S. V. Patankar, Numerical heat transfer and fluid flow, Boca Raton: CRC Press, 1980. https://doi.org/10.1201/9781482234213
[7]	J. Guermond, P. Minev, J. Shen, An overview of projection methods for incompressible flows, Comput. Method. Appl. M., 195 (2006), 6011–6045. https://doi.org/10.1016/j.cma.2005.10.010 doi: 10.1016/j.cma.2005.10.010
[8]	D. Kurmanova, Numerical modeling of coolant parameters in helicoid heat exchangers, Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, Computer Science, Mechanics Series, 146 (2024), 17–24. https://doi.org/10.32523/2616-7182/bulmathenu.2024/1.2 doi: 10.32523/2616-7182/bulmathenu.2024/1.2
[9]	J. Guermond, A. Salgado, A fractional step method based on a pressure Poisson equation for incompressible flows with variable density, C. R. Acad. Sci. Paris, Ser., 346 (2008), 913–918. https://doi.org/10.1016/j.crma.2008.06.006 doi: 10.1016/j.crma.2008.06.006
[10]	T. Tezduyar, S. Mittal, S. Ray, R. Shih, Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. Method. Appl. M., 95 (1992), 221–242. https://doi.org/10.1016/0045-7825(92)90141-6 doi: 10.1016/0045-7825(92)90141-6
[11]	P. Beisebay, A. Aruova, E. Akzhigitov, M. Tilepiev, About one splitting scheme for the nonlinear problem of thermal convection, Int. J. Heat Mass Trans., 104 (2017), 260–266. https://doi.org/10.1016/j.ijheatmasstransfer.2016.08.050 doi: 10.1016/j.ijheatmasstransfer.2016.08.050
[12]	G. Barrenechea, E. Castillo, D. Pacheco, Implicit-explicit schemes for incompressible flow problems with variable viscosity, SIAM J. Sci. Comput., 46 (2024), A2660–A2682. https://doi.org/10.1137/23M1606526 doi: 10.1137/23M1606526
[13]	D. Pacheco, R. Schussnig, Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions, Comput. Mech., 70 (2022), 977–992. https://doi.org/10.1007/s00466-022-02190-x doi: 10.1007/s00466-022-02190-x
[14]	M. Li, Y. Cheng, J. Shen, X. Zhang, A bound-preserving high-order scheme for variable-density incompressible Navier-Stokes equations, J. Comput. Phys., 425 (2021), 109906. https://doi.org/10.1016/j.jcp.2020.109906 doi: 10.1016/j.jcp.2020.109906
[15]	N. Li, J. Wu, X. Feng, Filtered time-stepping method for incompressible Navier-Stokes equations with variable density, J. Comput. Phys., 473 (2023), 111764. https://doi.org/10.1016/j.jcp.2022.111764 doi: 10.1016/j.jcp.2022.111764
[16]	W. Cai, B. Li, Y. Li, Error analysis of a fully discrete finite element method for variable density incompressible flows, ESAIM: M2AN, 55 (2021), S103–S147. https://doi.org/10.1051/m2an/2020029 doi: 10.1051/m2an/2020029
[17]	A. A. Samarskii, E. S. Nikolaev, Methods for solving grid equations (Russian), Moscow: Nauka, 1978.
[18]	G. de Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fl., 3 (1983), 249–264. https://doi.org/10.1002/fld.1650030305 doi: 10.1002/fld.1650030305
[19]	T. Nauryz, S. Kharin, A. Briozzo, J. Bollati, Exact solution to a Stefan-type problem for a generalized heat equation with the Thomson effect, Eurasian Math. J., 16 (2025), 68–89.
[20]	S. Kassabek, S. Kharin, D. Suragan, Exact and approximate solutions to the Stefan problem in ellipsoidal coordinates, Eurasian Math. J., 13 (2022), 51–66.
[21]	S. Jumabayev, D. Nurakhmetov, A. Aniyarov, Symmetry equivalences of boundary value problems for the nonuniform beams, J. Math. Mech. Comput. Sci., 126 (2025), 69–79. https://doi.org/10.26577/JMMCS2025126206 doi: 10.26577/JMMCS2025126206
[22]	A. Aniyarov, S. Jumabayev, D. Nurakhmetov, R. Kussainov, A hybrid algorithm for solving inverse boundary problems with respect to intermediate masses on a beam, Bull. Karaganda Univ.-Math., 4 (2021), 4–13. https://doi.org/10.31489/2021m4/4-13 doi: 10.31489/2021m4/4-13
[23]	S. Buranay, N. Arshad, Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method, Bull. Karaganda Univ.-Math., 1 (2023), 58–80. https://doi.org/10.31489/2023m1/58-80 doi: 10.31489/2023m1/58-80

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)