Geodesic
A method of solving grid equations for hydrodynamic problems in flat areas
Matematičeskoe modelirovanie, Tome 35 (2023) no. 3, pp. 35-58

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The paper discusses the numerical implementation of the mathematical model of the hydrodynamic process in the computational domain with "extended geometry", when its characteristic dimensions in the horizontal direction significantly exceed the vertical dimension. This is a typical property of a shallow water body or coastal system, which necessitates the development of specialized solution methods that arise in the process of discretization of grid equations. When solving the problem of transport in a shallow water body, the explicit-implicit scheme showed its effectiveness. The transition between time layers can be considered as an iterative process for solving the problem of diffusionconvection to settle. This idea formed the basis for the formation of a preconditioner in the proposed method for solving grid equations obtained by approximating hydrodynamic problems in areas with “extended geometry”. A numerical experiment was carried out with the developed software module, which made it possible to estimate the norm of the residual vector obtained by solving the grid equations of the pressure calculation problem based on the MPTM and the method for solving grid equations with a tridiagonal preconditioner, taking into account the hydrostatic approximation. According to the specifics of the developed method, it is effective in solving problems of aquatic ecology in the case of the computational domain, when its horizontal dimensions significantly exceed the vertical dimensions.
Keywords: mathematical modeling of hydrodynamic processes, computational domain with "elongated geometry", grid equations, modified alternately triangular method, method for solving grid equations with a three-diagonal preconditioner.
Mots-clés : explicit scheme 
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     author = {A. I. Sukhinov and A. E. Chistyakov and A. V. Nikitina and A. M. Atayan and V. N. Litvinov},
     title = {A method of solving grid equations for hydrodynamic problems in flat areas},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {35--58},
     publisher = {mathdoc},
     volume = {35},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2023_35_3_a2/}
}
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A. I. Sukhinov; A. E. Chistyakov; A. V. Nikitina; A. M. Atayan; V. N. Litvinov. A method of solving grid equations for hydrodynamic problems in flat areas. Matematičeskoe modelirovanie, Tome 35 (2023) no. 3, pp. 35-58. http://geodesic.mathdoc.fr/item/MM_2023_35_3_a2/