vmaitComputational Mathematics and Information TechnologiesComputational Mathematics and Information Technologies2587-8999Донской государственный технический университет10.23947/2587-8999-2018-2-76-90vmait-71Research ArticleСтатьиПостроение и исследование корректности математической модели транспорта и осаждения взвесей с учетом изменений рельефа днаAbout correctness of the suspension transport and sedimentation model, taking into account bottom relief changeshttps://orcid.org/0000-0002-5875-1523СухиновАлександр ИвановичSukhinovAlexander Ivanovich

Сухинов Александр Иванович, Донской государственный технический университет (344000 Ростов-на-Дону, пл. Гагарина, д. 1), доктор физико-математических наук, профессор

Sukhinov Alexander Ivanovich, Don State Technical University (1st Gagarin Square, Rostov-on-Don, Russian Federation), Doctor of Science in Physics and Maths, Professor

sukhinov@gmail.comСидорякинаВалентина ВладимировнаSidoryakinaValentina Vladimirovna

Сидорякина Валентина Владимировна, Таганрогский институт им. А.П. Чехова (филиал) РГЭУ (РИНЭ) (347936 Таганрог, улица Инициативная, д. 48), кандидат физико-математических наук, доцент

Sidoryakina Valentina Vladimirovna, Taganrog Institute of A.P. Chekhov (branch) RSUE (Initiative Street, Taganrog, Russian Federation), Candidate of Science in Physics and Maths, Associate professor

cvv9@mail.ruДонской государственный технический университет (344000 Ростов-на-Дону, пл. Гагарина, д. 1)РоссияDon State Technical University (1st Gagarin Square, Rostov-on-Don, Russian Federation)Russian FederationТаганрогский институт им. А.П. Чехова (филиал) РГЭУ (РИНЭ) (347936 Таганрог, улица Инициативная, д. 48)РоссияTaganrog Institute of A.P. Chekhov (branch) RSUE (Initiative Street, Taganrog, Russian Federation)Russian Federation2018280320232271Copyright © Сухинов А.И., Сидорякина В.В., 20232023Сухинов А.И., Сидорякина В.В.Sukhinov A.I., Sidoryakina V.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.cmit-journal.ru/jour/article/view/71

Настоящая работа посвящена исследованию пространственно-трехмерной модели транспорта и осаждения взвеси в прибрежной зоне с учетом изменения рельефа дна. Модель учитывает следующие процессы: адвективный перенос, обусловленный движением водной среды, микротурбулентную диффузию и гравитационное осаждение частиц взвеси, а также изменение геометрии дна, вызванное осаждением частиц взвеси или подъемом частиц донных отложений. Изменение рельефа дна приводит к необходимости решать начально-краевую задачу для уравнения параболического типа с младшими производными в области, геометрия которой зависит от искомой функции решения, что приводит, в общем случае, к нелинейной постановке задачи.Выполнена линеаризация модели на временной сетке за счет «замораживания» рельефа дна в пределах одного шага по времени и последующего пересчета функции поверхности дна на основе изменившейся функции концентрации взвешенного вещества, а также возможного изменения вектора скорости движения водной среды. Получена априорная оценка нормы решения в функциональном пространстве L2 в зависимости от интегральных оценок по времени правой части, граничных условий и начального условия, и, таким образом, доказана устойчивость решения исходной задачи от изменения начального и граничных условий и функции правой части. Модель может представлять ценность при прогнозе распространения загрязнений и изменения рельефа дна, как при антропогенном воздействии, так и в силу естественно протекающих природных процессов в прибрежной зоне.

This paper is devoted to the study of the spatial-three-dimensional model of sedimentation of suspended particles in the coastal zone, taking into account changes in the bottom topography. The model takes into account the following processes: advective transfer due to movement of the aquatic environment, microturbulent diffusion and gravitational sedimentation of suspended particles, as well as changes in bottom geometry caused by sedimentation of suspended particles or rise of sediment particles. A change in the bottom relief leads to the need to solve an initial-boundary value problem for an equation of parabolic type with lower derivatives in a domain whose geometry depends on the desired solution function, which leads, in general, to a non-linear formulation of the problem. The model was linearized on a time grid due to the «freezing» of the bottom relief within one time step and the subsequent recalculation of the bottom surface function based on the changed function of suspended matter concentration, as well as a possible change in the velocity vector of the aquatic environment. For the linearized problem, a quadratic functional was constructed and the energy method proved the uniqueness of the solution of the corresponding initial-boundary problem within an arbitrary time step. Based on the transformation of the quadratic functional, an a priori estimate of the solution norm in the L2 functional space was obtained depending on the integral estimates for the right side and the initial condition, and, thus, the stability of the solution of the initial problem against the change of the initial and boundary conditions and functions of the right side. The model may be of value in predicting the spread of pollution and changes in the bottom topography, both under anthropogenic impact and due to naturally occurring natural processes in the coastal zone.

прибрежные системыматематическая модельдиффузионно-конвективные задачи осаждения взвесиизменение рельефа днаединственность решения и устойчивость начально-краевой задачиcoastal systemsmathematical modeldiffusion-convection problems of deposition of suspended matterchange of the bottom reliefuniqueness of the solution and stability of the initial-boundary value problemРабота выполнена по теме № 2.6905.2017/БЧ в рамках выполнения госзадания Минобрнауки России в части НИР.The research is done on theme no. 6905.2017/БЧ within the frame of the government task of the Ministry of Education and Science of the Russian Federation in R&D.ReferencesSukhinov, A.I., Sidoryakina, V.V., Sukhinov, A.A.: Sufficient convergence conditionsSukhinov, A.I., Sidoryakina, V.V., Sukhinov, A.A.: Sufficient convergence conditionsfor positive solutions of linearized two-dimensional sediment transport problem. J. Computationalfor positive solutions of linearized two-dimensional sediment transport problem. J. 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The authors declare that there are no conflicts of interest present.