КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ НА ОСНОВІ НЕЛОКАЛЬНОЇ МОДЕЛІ ДИНАМІКИ КОНВЕКТИВНОЇ ДИФУЗІЇ РОЗЧИННИХ РЕЧОВИН У ПІДЗЕМНОМУ ФІЛЬТРАЦІЙНОМУ ПОТОЦІ В УМОВАХ МАСООБМІНУ

• Datum: 2025
• Hauptverfasser: Bohaienko, V.A., Bulavatsky, V.M.
• Format: Artikel
• Sprache: English
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Schlagworte: динаміка конвективно-дифузійних процесів; усталена плоско-вертикальна фільтрація ґрунтових вод; математичне і компʼютерне моделювання; дробово-диференціальні математичні моделі; похідна дробового порядку Капуто–Катугампола; нелінійні крайові задачі; скічненно-різницеві розв’язки
• Online Zugang: https://jais.net.ua/index.php/files/article/view/651
• Назва журналу: Problems of Control and Informatics

The paper deals with the problem of modeling the dynamics of locally none-quilibrium in time process of soluble substances convective diffusion under the conditions of flat-vertical steady-state groundwater filtration with free surface taking into account the presence of interfacial mass transfer. The urgency of solving such problems is due, in particular, to the need for the development of measures for soil flashing, as well as desalination and purification of groundwater from pollutants. For mathematical modeling of the corresponding transfer processes in media with a property of temporal non-locality, the apparatus of fractional-order integro-differentiation is used in the paper. A corresponding nonlinear fractional differential mathematical model of the migration process has been developed with the involvement of Caputo–Katugampola generalized fractional order derivative of a function with respect to another function, which allows in a certain sense to control the modeling process. In this model, the nonequilibrium convection-diffusion processes in porous media are considered under the conditions when mass exchange with host rocks is present. For the proposed mathematical model, the formulation of the corresponding boundary value problem was carried out and a technique for its numerical solution was developed. This technique is based on a preliminary transition using the conformal mapping method from the physical flow domain to the domain of complex potential, which is canonical. The algorithm for approximate solution of the considered boundary value problem in the domain of complex potential is based on a linearized version of the locally one-dimensional difference scheme of A.А. Samarsky. The given results of computer simulations demonstrate that the value of the exponent in the Caputo–Katugampola derivative significantly affects the simulation results, giving both sub-diffusion and super-diffusion patterns of the distribution of concentration fields. Computational experiments also show that when mass exchange phenomenon is taken into account while modeling pollution propagation from water bodies to soil media, it leads to a delay in the development of concentration front in liquid phase. In the paper we draw conclusions regarding the influence of mathematical model’s parameters on the resulting picture of concentration fields formation.