On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations
UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional...
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| Дата: | 2024 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2024
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7273 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865793857542684672 |
|---|---|
| author | Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang |
| author_facet | Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang |
| author_institution_txt_mv | [
{
"author": "Le Minh Hieu",
"institution": "Department of Economics, University of Economics – The University of Danang, Vietnam"
},
{
"author": " Nguyen Huu Nguyen Xuan",
"institution": "Department of Research and International Cooperation, University of Economics – The University of Danang, Vietnam"
},
{
"author": "Dang Ngoc Hoang Thanh",
"institution": "Department of Information Technology, School of Business Information Technology, University of Economics, Ho Chi Minh city, Vietnam"
}
] |
| author_sort | Hieu, Le Minh |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:35:01Z |
| description | UDC 517.9
We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions  are directly approximated on a two-point stencil of the second order. |
| doi_str_mv | 10.3842/umzh.v76i1.7273 |
| first_indexed | 2026-03-24T03:32:09Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7273 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:09Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-72732024-06-19T00:35:01Z On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang Maximum principle, two-side estimates, monotone method, finite-difference scheme, multidimensional quasilinear parabolic equation, convergence, weakly couple system, scientific computing, regularization principle, convection-diffusion problem, third boundary value problem. UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions  are directly approximated on a two-point stencil of the second order. УДК 517.9 Про нестандартний принцип максимуму та його застосування для побудови монотонних скінченно-різницевих схем для багатовимірних квазілінійних параболічних рівнянь Розглянуто принцип максимуму різниці з вхідними даними змінного знака та його застосування для дослідження монотонності та збіжності скінченно-різницевих схем (СРС). Зокрема, розглянуто початково-крайову задачу Діріхле для багатовимірного квазілінійного параболічного рівняння з необмеженою нелінійністю. Побудовано безумовно монотонні лінеаризовані скінченно-різницеві схеми другого порядку точності на рівномірних сітках. Отримано двосторонні оцінки для сіткового розв'язку, які повністю узгоджуються з аналогічними оцінками для точного розв'язку. Ці оцінки використано для доведення збіжності СРС у сітковій $L_2$-нормі. Також наведено дослідження щодо побудови монотонних різницевих схем другого порядку для параболічного рівняння конвекції-дифузії з крайовою умовою третього роду та необмеженою нелінійністю без використання вихідного диференціального рівняння на межах області. Нашою метою є поєднання припущення про існування та єдиність гладкого розв'язку з принципом регуляризації. Граничні умови в цьому випадку безпосередньо апроксимуються на двоточковому трафареті другого порядку.  Institute of Mathematics, NAS of Ukraine 2024-02-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7273 10.3842/umzh.v76i1.7273 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 1 (2024); 132 - 146 Український математичний журнал; Том 76 № 1 (2024); 132 - 146 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7273/9684 Copyright (c) 2024 Le Minh Hieu, Nguyen Huu Nguyen Xuan, Dang Ngoc Hoang Thanh |
| spellingShingle | Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang Hieu, Le Minh Xuan, Nguyen Huu Nguyen Thanh, Dang Ngoc Hoang On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_alt | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_full | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_fullStr | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_full_unstemmed | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_short | On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| title_sort | on the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations |
| topic_facet | Maximum principle two-side estimates monotone method finite-difference scheme multidimensional quasilinear parabolic equation convergence weakly couple system scientific computing regularization principle convection-diffusion problem third boundary value problem. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7273 |
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