ON A METHOD FOR CONSTRUCTING AN IRREGULAR GRID FOR THE ONE-DIMENSIONAL CONVECTION-DIFFUSION EQUATION
https://doi.org/10.26583/vestnik.2024.5.3
EDN: KWHCTL
Abstract
In this paper, we propose a new method for constructing an irregular grid for the numerical solution of problems containing a one-dimensional convection-diffusion equation, which is often encountered in various fields of computational mathematics, physics, and chemistry. Traditional approaches either use regular grids with a large number of nodes or adaptive grids that require rebuilding at each solution step, which can be computationally expensive. Our method is based on transforming a non-uniform grid into a uniform one using a local deformation function determined based on a monotonicity criterion. This allows us to obtain a monotonic solution on a grid with a significantly smaller number of nodes, thereby increasing the efficiency of the difference scheme. We consider both stationary and non-stationary convection-diffusion equations, describing the corresponding grid construction algorithms for divergent and non-divergent forms of recording convective terms. Examples of applying the method to various problems are given, demonstrating its advantages over existing approaches on regular grids. The presented approach combines the advantages of irregular grids to improve the solution efficiency and the use of a monotonicity criterion to ensure the stability of the scheme, expanding the capabilities of numerical methods for differential equations.
Supporting agencies: The study was supported by the Russian Science Foundation grant No. 23-41-00070, https://rscf.ru/project/23-41-00070/
About the Authors
S. A. Ladygin
National Research Nuclear University «MEPhI», Department of Applied Mathematics
Russian Federation
Russian Federation
R. N. Karachurin
National Research Nuclear University «MEPhI», Department of Applied Mathematics
Russian Federation
Russian Federation
K. E. Shilnikov
National Research Nuclear University «MEPhI», Department of Applied Mathematics;
Moscow Institute of Physics and Technology (MIPT), Department of Computational Physics
Russian Federation
Russian Federation
P. N. Ryabov
National Research Nuclear University «MEPhI», Department of Applied Mathematics
Russian Federation
Russian Federation
References
1. Godunov S.K., Ryaben’kii V.S. Raznostnye skhemy (vvedenie v teoriyu). Ucheb. pos., [Difference schemes (introduction to theory). Textbook]. Moscow, Nauka Publ., 1997.
2. Zenkevich O., Morgan K. Konechnye elementy i approksimatsiya [Finite Elements and Approximation]. Moscow, Mir Publ., 1986. 318 p. (in Russian).
3. Samarskii A.A., Vabishchevich P.N. Raznostnye metody resheniya zadach matematicheskoi fiziki na neregulyarnykh setkakh [Difference methods for solving problems of mathematical physics on irregular grids]. Matematicheskoe modelirovanie, 2001. Vol. 13. No. 2. Pp. 5–16 (in Russian).
4. Ladygin S.A., Karachurin R.N., Ryabov P.N., Kudryashov N.A. On Specifi Features of an Approach Based on Feedforward Neural Networks to Solve Problems Based on Differential Equations // Physics of Atomic Nuclei, 2023. Vol. 86. No. 10. Pp. 2231–2240.
5. Karachurin R.N., Ladygin S.A., Ryabov P.N., Shilnikov K.E., Kudryashov N.A. Exploring the Efficiency of Neural Networks for Solving Dynamic Process Problems: The Fisher Equation Investigation. Biologically Inspired Cognitive Architectures Meeting. Cham: Springer Nature Switzerland, 2023. Pp. 504–511.
6. Linß T. Layer-adapted meshes for convection-diffusion problems.Computer Methods in Applied Mechanics and Engineering, 2003. Vol. 192. No. 9–10. Pp. 1061–1105.
7. Roos H.G. Robust numerical methods for singularly perturbed differentia equations: a survey covering 2008–2012. International Scholarly Research Notices, 2012. Vol. 2012. No. 1. Pp. 379547.
8. Kuzmin D., Turek S. Flux correction tools for finite elements. Journal of Computational Physics, 2002. Vol. 175. No. 2. Pp. 525–558.
9. Mazhukin A.V., Mazhukin V.I. Dinamicheskaya adaptatsiya v parabolich-eskikh uravneniyakh [Dynamic adaptation in parabolic equations]. Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki, 2007. Vol. 47. No. 11. Pp. 1913–1936 (in Russian).
10. Shilnikov K.E., Kochanov M.B. On one approach for the numerical solving ofhyperbolic initial-boundary problems on an adaptive moving grids. Journal of Computational and Applied Mathematics, 2023. Vol. 421. Pp. 114884.
11. Shilnikov K.E., Kochanov M.B. Numerical solution of two-dimensional (2D) nonlinear heat conductivity problem on moving grids. Journal of Physics: Conference Series, 2020. Vol. 1686. No. 1. Pp. 012038.
12. Samarskii A.A., Vabishchevich P.N. Chislennye metody resheniya zadachkonvektsii-diffuzii [Numerical methods for solving convection-diffusion problems], Moscow, URSS Publ., 2003. 246 p.
13. Samarskii A.A., Gulin A.V. Chislennye metody. Uchebnoje posobie dlya vuzov [Numerical methods. Textbook, manual for universities]. Moscow, Nauka Publ., 1989. 432 p.
14. Chan R.H., Guo Y.Z., Lee S.T., Li X. Black–Scholes–Merton Model for Option Pricing. Financial Mathematics, Derivatives and Structured Products. Singapore, Springer Nature Singapore, 2024. Pp. 155–171.
15. Risken H., Risken H. Fokker-planck equation. Springer Berlin Heidelberg, 1996. Pp. 63–95.
16. Kolmogorov A. Ber die analytischen Methoden in der Wahrscheinlichkeits theorie. Math Annal, 1931. Vol. 104. Pp. 415–458.
Review
For citations:
Ladygin S.A., Karachurin R.N., Shilnikov K.E., Ryabov P.N. ON A METHOD FOR CONSTRUCTING AN IRREGULAR GRID FOR THE ONE-DIMENSIONAL CONVECTION-DIFFUSION EQUATION. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2024;13(5):303-315. (In Russ.) https://doi.org/10.26583/vestnik.2024.5.3. EDN: KWHCTL
Views:
128
Email this article 