Математичне моделювання повільного руху води у водоймі з ділянками поповнення течії методом комплексного аналізу

• Date: 2026
• Keywords: keywords
• Main Authors: Бомба, Андрій, Кушнір, Олекса
• Format: Article
• Language: Ukrainian
• Published: Kamianets-Podilskyi National Ivan Ohiienko University 2026
• Online Access: https://mcm-tech.kpnu.edu.ua/article/view/352179
• Journal Title: Mathematical and computer modelling. Series: Technical sciences

The study is devoted to the generalization of the numerical method of complex analysis of modeling processes of slow water movement in water bodies, limited by inflow, outflow and coastal impermeable lines. Namely, the planar problem of slow water movement in a water body with an arbitrary finite number of flow replenishment sections and one outflow section is considered, under the condition of «absence of overflows» (between the replenishment sections). To solve this problem, a generalization of the known numerical method of complex analysis is proposed, which is based on the construction of a conformal mapping of a given physical flow region onto the region of the corresponding complex potential, which has the form of a polygon with sides parallel to the coordinate axes. For this purpose, a complex flow potential consisting of the velocity potential and the flow function is introduced, and a corresponding inverse problem is formulated (for the conformal mapping of the complex potential domain onto the physical domain) with unknown geometric and hydrodynamic parameters, in particular the flows of each replenishment source and the total flow through the outflow section. The corresponding difference scheme is constructed based on the approximation of the Laplace equations and the Cauchy–Riemann conditions using «cross» and «T» type templates, and an iterative algorithm for refining the coordinates of the hydrodynamic grid nodes, conformality parameters, and flows is generalized. This algorithm is implemented programmatically in Python and tested on a test problem for a circular domain with two replenishment sources and one outflow source. A number of numerical experiments are carried out for cases of identical and different potential values in the replenishment sections. Hydrodynamic grids and velocity fields were constructed, and characteristic subregions of grid thickening and stagnation zones were identified.