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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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/

[1] A. A. Samarskii, P. N. Vabishchevich, “Finite-difference approximations to the transport equation. II”, Differ. Eq., 36:7 (2000), 1069–1077 | DOI | MR | MR

[2] A. A. Samarskij, P. N. Vabishhevich, Chislennye metody resheniia zadach konvektcii-diffuzii, URSS, M., 2009, 248 pp.

[3] A. A. Samarskiy, E. S. Nikolaev, Metody reshenija setochnyh uravnenij, Nauka, M., 1978

[4] A. N. Konovalov, “To the theory of the alternating triangle iteration method”, Siberian Math. J., 43:3 (2022), 439–457 | DOI | MR | MR

[5] B. N. Chetverushkin, “Kinetic models for supercomputer simulation continuous mechanic problems”, Math. Models Comput. Simul., 7:6 (2015), 531–539 | DOI | MR

[6] Y. L. Gurieva, V. P. Il'in, “O metodah soprjazhennyh napravlenij dlja mnogokratnogo reshenija SLAU”, Chislennye metody i voprosy organizacii vychislenij Part XXXIII, Zap. POMI, 496, 2020, 26–42

[7] M. V. Yakobovskiy, S. K. Grigorjev, “Algoritm garantirovannoj generacii tetrajedral'noj setki proekcionnym metodom”, Keldysh Institute preprints, 2018, 109, 18 pp.

[8] N. L. Zamarashkin, I. V. Oseledets, E E. Tyrtyshnikov, “New applications of matrix methods”, Comput. Math. Math. Phys., 61:5 (2021), 669–673 | DOI | MR | MR

[9] Yu. V. Vassilevski, S. S. Simakov et al., “Personalization of mathematical models in cardiology: obstacles and perspectives”, Comput. Res. Mod., 14:4 (2022), 911–930

[10] O. Yu. Milyukova, “MPI+OpenMP realizacija metoda soprjazhennyh gradientov s predobuslovlivatelem blochnogo nepolnogo obratnogo treugol'nogo razlozhenija pervogo porjadka”, Vych. met. programmirovanie, 23:3 (2022), 191–206 | MR

[11] J. H. Bramble, J. E. Pasciak, “Analysis of a Cartesian PML Approximation to the Three Dimensional Electromagnetic Scattering Problem”, Intern. J. Numer. Anal. Model., 9:3 (2012), 543–561 | MR

[12] A. Kleefeld, L. Tzu-Chu, “Boundary Element Collocation Method for Solving the Exterior Neumann Problem for Helmholtz-sEquation in Three Dimensions”, Electron. Trans. Numer. Anal., 2012, no. 39, 113–143 | MR

[13] A. I. Sukhinov, A. E. Chistiakov, V. V. Sidoriakina, E. A. Protsenko, “Jekonomichnye javnonejavnye shemy reshenija mnogomernyh zadach diffuzii-konvekcii”, Vychisl. mehanika sploshnyh sred, 12:4 (2019), 435–445

[14] A. I. Sukhinov, A. E. Chistyakov, E. V. Alekseenko, “Numerical realization of three-dimensional model of hydrodynamics for shallow water basins on high-performance system”, Math. Models Comput. Simul., 3:5 (2011), 562–574 | DOI | MR

[15] E. Alekseenko, B. Roux, D. Fougere, P. G. Chen, “The Effect of Wind Induced Bottom Shear Stress and Salinity on Zostera Noltii Replanting in A Mediterranean Coastal Lagoon”, Estuarine, Coastal and Shelf Science, 187 (2017), 293–305 | DOI

[16] O. M. Belotserkovskii, V. A. Gushchin, V. V. Shchennikov, “Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid”, Comput. Math. Math. Phys., 15:1 (1975), 190–200 | DOI | MR

[17] A. I. Sukhinov, A. E. Chistyakov, I. Y. Kuznetsova, A. M. Atayan, A. V. Nikitina, “Regularized difference scheme for solving hydrodynamic problems”, Math. Models Comput. Simul., 14:5 (2022), 745–754 | DOI | MR | MR

[18] A. I. Sukhinov, A. V. Nikitina, A. M. Atayan, V. N. Litvinov, Yu. V. Belova, A. E. Chistyakov, “Supercomputer simulation of hydrobiological processes of coastal systems”, Math. Models Comput. Simul., 14:4 (2022), 677–690 | DOI | MR | MR

[19] A. I. Sukhinov, A. E. Chistyakov, E. A. Protsenko, V. V. Sidoryakina, S.V Protsenko, “Accounting Method of Filling Cells for the Solution of Hydrodynamics Problems with a Complex Geometry of the Computational Domain”, MMCS, 12:2 (2020), 232–245 | MR

[20] L. V. Kantorovich, “Funkcional'nyj analiz i prikladnaja matematika”, Uspekhi Mat. Nauk, 3:6 (1948), 89–185 | MR

[21] B. N. Chetverushkin, M. V. Yakobovskiy, M. A. Kornilina, A. V. Semenova, “Numerical algorithms for HPC systems and fault tolerance”, Comm. in Comput. Infor.Sci., 1063 (2019), 34–44

[22] N. D'Ascenzo, B. N. Chetverushkin, V. I., “Saveliev On an algorithm for solving parabolic and elliptic equations”, Comput. Math. Math. Phys., 55:8 (2015), 1290–1297 | DOI | MR