A monotone finite-difference high order accuracy scheme for the 2D convection – diffusion equations

DOI: https://doi.org/10.33581/2520-6508-2019-3-71-83

Abstract

A stable finite-difference scheme is constructed on a minimum stencil of a uniform mesh for a two-dimensional steady-state convection – diffusion equation of a general form; the scheme is theoretically studied and tested. It satisfies the maximum principle and has the fourth order of approximation. The scheme monotonicity is controlled by two regularization parameters introduced into the difference operator. The scheme is focused on solving applied convection – diffusion problems with a developed boundary layer, including gravitational convection, thermomagnetic convection, and diffusion of particles in a magnetic fluid. The scheme is tested on the well-known problem of a high-intensive gravitational convection in a horizontal channel of a square cross-section with a uniform heating from the side. A detailed comparison is performed with the monotone Samarskii scheme of the second order approximation on the sequences of square meshes with the number of partitions from 10 to 1000 on each side of the square domain and over the entire range of the Rayleigh numbers, corresponding to the laminar convection mode. A significant advantage of the fourth order scheme in the convergence rate is shown for the decreasing mesh step.

Author Biography

Viktor K. Polevikov, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; associate professor at the department of computational mathematics, faculty of applied mathematics and computer science

References

  1. Beresnev S, Polevikov V, Tobiska L. Numerical study of the influence of diffusion of magnetic particles on equilibrium shapes of a free magnetic fluid surface. Communications in Nonlinear Science and Simulation. 2009;14(4):1403–1409. DOI: 10.1016/j.cnsns.2008.04.005.
  2. Polevikov V, Tobiska L. Influence of diffusion of magnetic particles on stability of a static magnetic fluid seal under the action of external pressure drop. Communications in Nonlinear Science and Numerical Simulation. 2011;16(10):4021– 4027. DOI: 10.1016/j. cnsns.2011.02.025.
  3. Samarskii AA. The theory of difference schemes. New York: Marcel Dekker; 2001. 73 p.
  4. Roos H-G, Stynes M, Tobiska L. Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems. Berlin: Springer; 2008. 604 p. (Springer Series in Computational Mathematics; volume 24). DOI: 10.1007/978-3-540-34467-4.
  5. Stynes M, Stynes D. Convection-diffusion problems: an introduction to their analysis and numerical solution. USA: American Mathematical Society; 2018. 156 p. (Graduate studies in mathematics; volume 196). Co-published by Atlantic Association for Research in the Mathematical Sciences.
  6. Samarskii AA, Vabishchevich PN. Chislennye metody resheniya zadach konvektsii-diffuzii [Numerical methods for solving convection-diffusion problems]. 4th edition. Moscow: Librokom; 2009. 248 p. Russian.
  7. Lemeshevsky SV, Matus PP, Yakubuk RM. Two-layered higher-order difference schemes for the convection-diffusion equation. Doklady of the National Academy of Sciences of Belarus. 2012;56(2):15–18. Russian.
  8. Berkovski B, Bashtovoi V, editors. Magnetic fluids and applications handbook. New York: Begell House; 1996. 831 p. (UNESCO series of learning materials).
  9. Berkovsky BM, Medvedev VF, Krakov MS. Magnetic fluids: engineering applications. New York: Oxford University Press; 1993. 243 p.
  10. Fertman VE. Magnitnye zhidkosti – estestvennaya konvektsiya i teploobmen [Magnetic fluids: natural convection and heat transfer]. Minsk: Nauka i tekhnika; 1978. 205 p. Russian.
  11. Bashtovoi VG, Berkovsky BM, Vislovich AN. Introduction to thermomechanics of magnetic fluids. Washington: Hemisphere Publishing; 1988. 228 p.
  12. Berkovsky BM, Polevikov VK. Vychislitel’nyi eksperiment v konvektsii [Computational experiment in convection]. Minsk: Universitetskoe; 1988. 167 p. Russian.
  13. Polevikov V, Tobiska L. On the solution of the steady-state diffusion problem for ferromagnetic particles in a magnetic fluid. Mathematical Modeling and Analysis. 2008;13(2):233–240. DOI: 10.3846/1392-6292.2008.13.233-240.
  14. Berkovskii BM, Polevikov VK. Effect of the Prandtl number on the convection field and the heat transfer during natural convection. Journal of Engineering Physics. 1973;24(5):598 – 603. DOI: 10.1007/BF00838619.
  15. Gershuni GZ, Zhukhovitskii EM, Tarunin EL. Numerical investigation of convective motion in a closed cavity. Fluid Dynamics. 1966;1(5):38 – 42. DOI: 10.1007/BF01022148.
  16. Polevikov VK. [The difference scheme of fourth order accuracy to compute the boundary vorticity in hydrodynamics problems]. Doklady Akademii nauk Belorusskoi SSR. 1979;23(10):872–875. Russian.
  17. Polevikov VK. Application of the relaxation method to solve steady difference problems of convection. USSR Computational Mathematics and Mathematical Physics. 1981;21(1):126 –137. DOI: 10.1016/0041-5553(81)90138-5.
  18. Gosman AD, Pun WM, Runchal AK, Spalding DB, Wolfshtein M. Heat and mass transfer in recirculating flows. London: Academic Press; 1969. 338 p.
Published
2019-11-27
Keywords: gravitational convection, thermomagnetic convection, diffusion of particles, diffusion-convection equation, scheme of high accuracy, parameters of regularization, monotonicity, test problem
How to Cite
Polevikov, V. K. (2019). A monotone finite-difference high order accuracy scheme for the 2D convection – diffusion equations. Journal of the Belarusian State University. Mathematics and Informatics, 3, 71-83. https://doi.org/10.33581/2520-6508-2019-3-71-83
Issue
No 3 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Computational Mathematics

Copyright (c) 2019 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)