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).&nbsp;Namely, we consider the Dirichlet initial-boundary value problem for multidimensional...

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Date:	2024
Main Authors:	Hieu, Le Minh, Xuan, Nguyen Huu Nguyen, Thanh, Dang Ngoc Hoang
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Language:	English
Published:	Institute of Mathematics, NAS of Ukraine 2024
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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_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).&nbsp;Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity.&nbsp;Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids.&nbsp;A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained.&nbsp;These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm.&nbsp;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.&nbsp;The goal is a combination of the assumption of existence and&nbsp;uniqueness of a smooth solution and the regularization principle.&nbsp;In this case, the boundary conditions&nbsp; 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	Skip to main content Log in Menu Find a journal Publish with us Track your research Search Saved research Cart Home Ukrainian Mathematical Journal Article On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations Published: 30 July 2024 Volume 76, pages 141–156, (2024) Cite this article Save article View saved research Ukrainian Mathematical Journal Aims and scope Submit manuscript Le Minh Hieu1, Nguyen Huu Nguyen Xuan2 & Dang Ngoc Hoang Thanh3 45 Accesses 1 Citation Explore all metrics 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 equations with 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 L2-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. This is a preview of subscription content, log in via an institution to check access. 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Institutional subscriptions Similar content being viewed by others Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind Article 01 July 2021 Monotone Finite-Difference Schemes of Second-Order Accuracy for Quasilinear Parabolic Equations with Mixed Derivatives Article 01 March 2019 Second Order Monotone Difference Schemes with Approximation on Non-Uniform Grids for Two-Dimensional Quasilinear Parabolic Convection-Diffusion Equations Article 01 April 2020 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Computational Mathematics and Numerical Analysis Ordinary Differential Equations Partial Differential Equations Functional Analysis Partial Differential Equations on Manifolds Calculus of Variations and Optimization Functional Analysis and Boundary Value Problems in Differential Equations References V. 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Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1964). Download references Author information Authors and Affiliations Department of Economics, University of Economics, The University of Danang, Danang, Vietnam Le Minh Hieu Department of Research and International Cooperation, University of Economics, The University of Danang, Danang, Vietnam Nguyen Huu Nguyen Xuan Department of Information Technology, School of Business Information Technology, University of Economics, Ho Chi Minh City, Vietnam Dang Ngoc Hoang Thanh Authors Le Minh HieuView author publications Search author on:PubMed Google Scholar Nguyen Huu Nguyen XuanView author publications Search author on:PubMed Google Scholar Dang Ngoc Hoang ThanhView author publications Search author on:PubMed Google Scholar Corresponding author Correspondence to Le Minh Hieu. Additional information Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 132–146, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.7273. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and permissions About this article Cite this article 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. Ukr Math J 76, 141–156 (2024). https://doi.org/10.1007/s11253-024-02313-y Download citation Received: 29 July 2022 Published: 30 July 2024 Version of record: 30 July 2024 Issue date: June 2024 DOI: https://doi.org/10.1007/s11253-024-02313-y Share this article Anyone you share the following link with will be able to read this content: Get shareable linkSorry, a shareable link is not currently available for this article. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Profiles Nguyen Huu Nguyen Xuan View author profile Dang Ngoc Hoang Thanh View author profile Access this article Log in via an institution Subscribe and save Springer+ from €37.37 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF 39,95 € Price includes VAT (Ukraine) Instant access to the full article PDF. 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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).&nbsp;Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity.&nbsp;Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids.&nbsp;A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained.&nbsp;These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm.&nbsp;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.&nbsp;The goal is a combination of the assumption of existence and&nbsp;uniqueness of a smooth solution and the regularization principle.&nbsp;In this case, the boundary conditions&nbsp; 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).&nbsp;Зокрема, розглянуто початково-крайову задачу Діріхле для багатовимірного квазілінійного параболічного рівняння з необмеженою нелінійністю.&nbsp;Побудовано безумовно монотонні лінеаризовані скінченно-різницеві схеми другого порядку точності на рівномірних сітках.&nbsp;Отримано двосторонні оцінки для сіткового розв'язку, які повністю узгоджуються з аналогічними оцінками для точного розв'язку.&nbsp;Ці оцінки використано для доведення збіжності СРС у сітковій $L_2$-нормі.&nbsp;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.&nbsp;Нашою метою є поєднання припущення про існування та єдиність гладкого розв'язку з принципом регуляризації.&nbsp;Граничні умови в цьому випадку безпосередньо апроксимуються на двоточковому трафареті другого порядку.&nbsp; 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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On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

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