Monotone difference schemes for quasilinear parabolic equations with reaction coefficients of arbitrary sign
https://doi.org/10.29235/1561-8323-2026-70-3-183-190
Abstract
This paper establishes continuous and discrete two-sided estimates for a class of quasilinear parabolic convection–diffusion–reaction equations with unbounded nonlinearities and reaction coefficients of arbitrary sign. For nonpositive reaction coefficients, rigorous continuous two-sided bounds are derived for classical solutions, together with an a priori estimate in the strong C-norm. These results extend, in a structurally coherent manner, the corresponding theory available for linear parabolic problems. For reaction coefficients of arbitrary sign, second-order unconditionally monotone schemes are constructed. By applying the Ladyzhenskaya transformation u = veλt, alternative discrete two-sided estimates are derived for the numerical solution and shown to be fully consistent with their continuous counterparts. Finally, convergence of the developed schemes is rigorously proved in the L2-norm.
About the Authors
P. P. Matus
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Matus Piotr P. – Corresponding Member, D. Sc. (Physics and Mathematics), Professor, Chief Researcher
Minsk
Hieu Le Minh
University of Economics, The University of Danang
Viet Nam
Viet Nam
Le Minh Hieu – Ph. D. (Physics and Mathematics), Lecturer, Researcher
71, Ngu Hanh Son Str., 50000, Da Nang
A. L. Gladkov
Belarusian State University
Belarus
Belarus
Gladkov Alexander L. – D. Sc. (Physics and Mathem atics), Professor, Head of the Department
Minsk
References
1. Ladyzhenskaya O. A., Solonnikov V. A., Ural’tseva, N. N. Linear and Quasilinear Equations of Parabolic Type. Moscow, 1967. 736 p. (in Russian).
2. Friedman A. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964. 347 p.
3. Farago I., Horvath R. Discrete maximum principle and adequate discretizations of linear parabolic problems. SIAM Journal on Scientific Computing, 2006, vol. 28, no. 6, pp. 2313–2336. https://doi.org/10.1137/050627241
4. Fujii H. Some remarks on finite element analysis of time-dependent field problems. Theory and Practice of Finite Element Structural Analysis, 1973, pp. 91–106.
5. Keller H. B. The numerical solution of parabolic partial differential equations. Ralston A., Wilf H. S., eds. Mathematical Methods for Digital Computers. New York, 1960, pp. 135–143.
6. Stys T., Motsumi T., Daman O. On the maximum principle and its application to diffusion equations. Numerical Methods for Partial Differential Equations, 2007, vol. 23, no. 1, pp. 60–72. https://doi.org/10.1002/num.20168
7. Farago I., Karatson J., Korotov S. Discrete maximum principles for nonlinear parabolic PDE systems. IMA Journal of Numerical Analysis, 2012, vol. 32, no. 4, pp. 1541–1573. https://doi.org/10.1093/imanum/drr050
8. Samarskii A. A. The Theory of Difference Schemes. New York, 2001. 786 p. https://doi.org/10.1201/9780203908518
9. Samarskii A. A., Matus P. P., Vabishchevich P. N. Difference Schemes with Operator Factors. Dordrecht, 2002. https://doi.org/10.1007/978-94-015-9874-3
10. Matus P. P. The maximum principle and some of its applications. Computational Methods in Applied Mathematics, 2002, vol. 2, no. 1, pp. 50–91. https://doi.org/10.2478/cmam-2002-0004
11. Matus P. P., Gaspar F. J., Hieu L. M., Tuyen V. T. K. Monotone difference schemes for weakly coupled elliptic and parabolic systems. Computational Methods in Applied Mathematics, 2017, vol. 17, no. 2, pp. 287–298. https://doi.org/10.1515/cmam-2016-0046
12. Matus P. P., Hieu L. M., Vulkov L. G. Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations. Journal of Computational and Applied Mathematics, 2017, vol. 310, pp. 186–199. https://doi.org/10.1016/j.cam.2016.04.006
13. Hieu L. M., Xuan N. H. N., Thanh D. N. H. On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations. Ukrainian Mathematical Journal, 2024, vol. 76, no. 1, pp. 141–156. https://doi.org/10.1007/s11253-024-02313-y
14. Matus P. P., Hieu L. M., Pylak D. Monotone finite-difference schemes of second-order accuracy for quasilinear parabolic equations with mixed derivatives. Differential Equations, 2019, vol. 55, no. 3, pp. 424–436. https://doi.org/10.1134/ S0012266119030157
15. Matus P. P., Rybak I. V. Monotone difference schemes for nonlinear parabolic equations. Differential Equations, 2003, vol. 39, no. 7, pp. 1013–1022. https://doi.org/10.1023/B:DIEQ.0000009197.94879.5c
Review
Views:
1735
JATS XML
Email this article 