Numerical solution of the mixed boundary value problem for the heat equation in two-dimensional domains of complex shape
https://doi.org/10.29235/1561-2430-2024-60-3-216-224
Abstract
A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.
About the Authors
M M. Chuiko
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Mikhail M. Chuiko – Ph. D. (Physics and Mathematics), Leading Researcher of the Department of Computational Mathematics
11, Surganov Str., 220072, Minsk
O. M. Korolyova
Belarusian National Technical University
Belarus
Belarus
Olga M. Korolyova – Ph. D. (Physics and Mathematics), Associate Professor of the Department of Higher Mathematics
65, Nezalezhnosti Ave., 220013, Minsk
References
1. Ingram D. M., Causon D. M., Mingham C. G. Developments in cartesian cut cell methods. Mathematics and Computers in Simulation, 2003, vol. 61, no. 3–6, pp. 561–572. https://doi.org/10.1016/s0378-4754(02)00107-6
2. Kirkpatrick M. P., Armfield S. W., Kent J. H. A representation of curved boundaries for the solution of the Navier– Stokes equations on a staggered three-dimensional Cartesian grid. Journal of Computational Physics, 2003, vol. 184, no. 1, pp. 1–36. https://doi.org/10.1016/s0021-9991(02)00013-x
3. Ye T., Mittal R., Udaykumar H. S., Shyy W. An accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries. Journal of Computational Physics, 1999, vol. 156, no. 2, pp. 209–240. https://doi.org/10.1006/jcph.1999.6356
4. Mittal R., Iaccarino G. Immersed boundary methods. Annual Review of Fluid Mechanics, 2005, vol. 37, pp. 239–261. https://doi.org/10.1146/annurev.fluid.37.061903.175743
5. LeVeque R. J., Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis, 1994, vol. 31, no. 4, pp. 1019–1044. https://doi.org/10.1137/0731054
6. Li Z. An overview of the immersed interface method and its applications. Taiwanese Journal of Mathematics, 2003, vol. 7, no. 1, pp. 1–49. https://doi.org/10.11650/twjm/1500407515
7. Vabishchevich P. N. Method of Fictitious Domains in the Problems of Mathematical Physics. Moscow, Moscow University Publishing House, 1991. 156 p. (in Russian).
8. Vinnikov V. V., Reviznikov D. L. Cartesian grids methods for numerical solution of Navier-Stokes equations in domains with curvilinear boundaries. Matematicheskoe modelirovanie = Mathematical Models and Computer Simulation, 2005, vol. 17, no. 8, pp. 15–30 (in Russian).
9. Fletcher C. A. J. Computational Techniques for Fluid Dynamics. Vol. 2. Berlin, Heidelberg, Springer, 1988. 494 p. https://doi.org/10.1007/978-3-642-97071-9
10. Thompson J. F., Warsi Z. U. A., Mastin C. W. Numerical Grid Generation: Foundations and Applications. New York, Elsevier North-Holland, 1985. 483 p.
11. Mastin C. W. Error induced by coordinate systems. Applied Mathematics and Computation, 1982, vol. 10–11, pp. 31– 40. https://doi.org/10.1016/0096-3003(82)90186-2
12. Samarskii A. A. The Theory of Difference Schemes. New York, 2001. 786 p. https://doi.org/10.1201/9780203908518
13. Samarskii A. A., Andreev V. B. Difference Methods for Elliptic Equations. Moscow, Nauka Publ., 1976. 352 p. (in Russian).
14. Samarskii A., Matus P., Mazhukin V., Mozolevski I. Monotone difference schemes for equations with mixed derivatives. Computers & Mathematics with Applications, 2002, vol. 44, no. 3–4, pp. 501–510. https://doi.org/10.1016/S0898-1221(02)00164-5
15. Rybak I. Monotone and conservative difference schemes for elliptic equations with mixed derivatives. Mathematical Modelling and Analysis, 2004, vol. 9, pp. 169–178. https://doi.org/10.3846/13926292.2004.9637250
16. Matus P. P., Le Minh Hieu, Pylak D. Difference schemes for quasilinear parabolic equations with mixed derivatives. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2019, vol. 63, no. 3, pp. 263–269 (in Russian). https://doi.org/10.29235/1561-8323-2019-63-3-263-269
17. Gaspar F. J., Lisbona F. J., Matus P., Tuyen V. T. K. Monotone finite difference schemes for quasilinear parabolic problems with mixed boundary conditions. Computational Methods in Applied Mathematics, 2016, vol. 16, no. 2, pp. 231–244. https://doi.org/10.1515/cmam-2016-0002
18. Chuiko M. M., Korolyova O. M. Solution of the mixed boundary problem for the Poisson equation on two-dimensional irregular domains. Informatika = Informatics, 2023, vol. 20, no. 2, pp. 111−120 (in Russian). https://doi.org/10.37661/1816-0301-2023-20-2-111-120
19. Schneider G. E., Zedan M. A modified strongly implicit procedure for the numerical solution of field problem. Numerical Heat Transfer, 1981, vol. 4, no. 1, pp. 1–19. https://doi.org/10.1080/01495728108961775
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