danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2026-70-3-183-190dan-1308Research ArticleМАТЕМАТИКАMATHEMATICSМонотонные разностные схемы для квазилинейных параболических уравнений с коэффициентами реакции произвольного знакаMonotone difference schemes for quasilinear parabolic equations with reaction coefficients of arbitrary signМатусП. П.MatusP. P.

Матус Петр Павлович – д-р физ.-мат. наук, проф ессор, гл. науч. сотрудник

ул. Сурганова, 11, 220072, Минск

Matus Piotr P. – Corresponding Member, D. Sc. (Physics and Mathematics), Professor, Chief Researcher

Minsk

matus@im.bas-net.byЛе МиньХиеуLe MinhHieu

Ле Минь Хиеу – канд. физ.-мат. наук, преподаватель, научный сотрудник

ул. Нгу Хань Шон, 71, 550000, Дананг

Le Minh Hieu – Ph. D. (Physics and Mathematics), Lecturer, Researcher

71, Ngu Hanh Son Str., 50000, Da Nang

hieulm@due.edu.vnГладковА. Л.GladkovA. L.

Гладков Александр Львович – д-р физ.-мат. наук, профессор, заведующий кафедрой

пр. Независимости, 4, 220030, Минск

Gladkov Alexander L. – D. Sc. (Physics and Mathem atics), Professor, Head of the Department

Minsk

gladkoval@bsu.byИнститут математики Национальной академии наук БеларусиБеларусьInstitute of Mathematics of the National Academy of Sciences of BelarusBelarusУниверситет экономики, Университет ДанангаВьетнамUniversity of Economics, The University of DanangViet NamБелорусский государственный университетБеларусьBelarusian State UniversityBelarus202609072026703183190Copyright © Матус П.П., Ле Минь Х., Гладков А.Л., 20262026Матус П.П., Ле Минь Х., Гладков А.Л.Matus P.P., Le Minh H., Gladkov A.L.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1308

Установлены непрерывные и дискретные двусторонние оценки для класса квазилинейных параболических уравнений конвекции–диффузии–реакции с неограниченными нелинейностями и коэффициентами реакции произвольного знака. Для неположительных коэффициентов реакции выведены строгие непрерывные двусторонние оценки для классических решений, а также априорная оценка в сильной C-норме. Полученные результаты структурно непротиворечивым образом расширяют соответствующую теорию, доступную для линейных параболических задач. Для коэффициентов реакции произвольного знака построены безусловно монотонные схемы второго порядка. С помощью преобразования Ладыженской u = veλt выведены альтернативные дискретные двусторонние оценки для численного решения, которые, как показано, полностью согласуются с их непрерывными аналогами.

Сходимость разработанных схем строго доказана в L2-норме.

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.

принцип максимумадвусторонние оценкиквазилинейное параболическое уравнение конвекции–диффузии–реакциикоэффициент реакции произвольного знакасходимостьмонотонная разностная схемаmaximum principletwo-sided estimatesquasilinear parabolic convection–diffusion–reaction equationreaction coefficient of arbitrary signconvergencemonotone difference schemeReferencesLadyzhenskaya O. A., Solonnikov V. A., Ural’tseva, N. N. Linear and Quasilinear Equations of Parabolic Type. Moscow, 1967. 736 p. (in Russian).Ladyzhenskaya O. A., Solonnikov V. A., Ural’tseva, N. N. Linear and Quasilinear Equations of Parabolic Type. Moscow, 1967. 736 p. (in Russian).Friedman A. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964. 347 p.Friedman A. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964. 347 p.Farago I., Horvath R. Discrete maximum principle and adequate discretizations of linear parabolic problems. 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Computational Methods in Applied Mathematics, 2017, vol. 17, no. 2, pp. 287–298. https://doi.org/10.1515/cmam-2016-0046Matus 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-2016-0046Matus 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.2016.04.006Matus 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.2016.04.006Hieu 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-yHieu 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-yMatus 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/ S0012266119030157Matus P. P., Hieu L. M., Pylak D. 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The authors declare that there are no conflicts of interest present.