Abstract:
The paper discusses the development and application of the accounting rectangular cell fullness method with material substance, in particular, a liquid, to increase the smoothness and accuracy of a finite-difference solution of hydrodynamic problems with a complex shape of the boundary surface. Two problems of computational hydrodynamics are considered to study the possibilities of the proposed difference schemes: the spatial-two-dimensional flow of a viscous fluid between two coaxial semi-cylinders and the transfer of substances between coaxial semi-cylinders. Discretization of diffusion and convection operators was performed on the basis of the integro-interpolation method, taking into account the fullness of cells and without it. It is proposed to use a difference scheme, for solving the problem of diffusion–convection at large grid Peclet numbers, that takes into account the cell population function, and a scheme on the basis of linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error at small Courant numbers. As a reference, an analytical solution describing the Couette–Taylor flow is used to estimate the accuracy of the numerical solution. The relative error of calculations reaches 70 % in the case of the direct use of rectangular grids (stepwise approximation of the boundaries), under the same conditions using the proposed method allows to reduce the error to 6 %. It is shown that the fragmentation of a rectangular grid by 2–8 times in each of the spatial directions does not lead to the same increase in the accuracy that numerical solutions have, obtained taking into account the fullness of the cells. The proposed difference schemes on the basis of linear combination of the Upwind and Standard Leapfrog difference schemes with weighting factors of 2/3 and 1/3, respectively, obtained by minimizing the order of approximation error, for the diffusion–convection problem have a lower grid viscosity and, as a corollary, more precisely, describe the behavior of the solution in the case of large grid Peclet numbers.
Citation:
A. I. Sukhinov, A. E. Chistyakov, E. A. Protsenko, “Difference scheme for solving problems of hydrodynamics for large grid Peclet numbers”, Computer Research and Modeling, 11:5 (2019), 833–848
\Bibitem{SukChiPro19}
\by A.~I.~Sukhinov, A.~E.~Chistyakov, E.~A.~Protsenko
\paper Difference scheme for solving problems of hydrodynamics for large grid Peclet numbers
\jour Computer Research and Modeling
\yr 2019
\vol 11
\issue 5
\pages 833--848
\mathnet{http://mi.mathnet.ru/crm745}
\crossref{https://doi.org/10.20537/2076-7633-2019-11-5-833-848}
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https://www.mathnet.ru/eng/crm745
https://www.mathnet.ru/eng/crm/v11/i5/p833
This publication is cited in the following 11 articles:
N. V. Kudinov, A. A. Filina, A. V. Nikitina, D. V. Bondarenko, I. F. Razveeva, “Simulation of Vertical Movements of Seawater in Stratified Reservoirs”, Vestnik Donskogo gosudarstvennogo tehničeskogo universiteta, 23:2 (2023), 212
Abdulkhakim Salokhiddinov, Andre Savitsky, Maria Radkevich, Olga Ashirova, O. Tursunov, “Possibilities of solving two-dimensional hydrodynamic problems on the basis of the non-divergent form of recording the transport and conservation equations”, E3S Web Conf., 434 (2023), 01001
A. E. Chistyakov, A. V. Nikitina, I. Yu. Kuznetsova, E. O. Rakhimbaeva, M. V. Porksheyan, “Investigation of the Approximation Error of the Difference Scheme for the Mathematical Model of Hydrodynamics”, Lobachevskii J Math, 44:5 (2023), 1839
A. I. Sukhinov, A. E. Chistyakov, A. M. Atayan, I. Yu. Kuznetsova, V. N. Litvinov, A. V. Nikitina, “Matematicheskaya model protsessa osazhdeniya na dno mnogokomponentnoi vzvesi i izmeneniya sostava donnykh materialov”, Izv. IMI UdGU, 60 (2022), 73–89
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
Alexander Sukhinov, Alexander Chistyakov, Inna Kuznetsova, Yulia Belova, Alla Nikitina, “Mathematical Model of Suspended Particles Transport in the Estuary Area, Taking into Account the Aquatic Environment Movement”, Mathematics, 10:16 (2022), 2866
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
A E Chistyakov, E A Protsenko, I Y Kuznetsova, A V Nikitina, “Suspended particle transport process modeling based on 2D and 3D models”, J. Phys.: Conf. Ser., 1902:1 (2021), 012137
Asya M. Atayan, Inna Yu. Kuznetsova, Communications in Computer and Information Science, 1437, Parallel Computational Technologies, 2021, 312
A. I. Sukhinov, I. Yu. Kuznetsova, A. E. Chistyakov, E. A. Protsenko, Yu. V. Belova, “Studying the Accuracy and Applicability of the Finite Difference Scheme for Solving the Diffusion–Convection Problem at Large Grid Péclet Numbers”, J Appl Mech Tech Phy, 62:7 (2021), 1255
A.I. Sukhinov, I.Y. Kuznetsova, A.E. Chistyakov, E.A. Protsenko, Y.V. Belova, “Study of the accuracy and applicability of the difference scheme for solving the diffusion-convection problem at large grid Péclet numbers”, Comp. Contin. Mech., 13:4 (2020), 437
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