Solving the shallow water problem by central differences and FCT correction
https://doi.org/10.15514/ISPRAS-2022-34(5)-17
Abstract
The paper proposes an algorithm based on central differences and FCT correction for solving the shallow water problem. The results of numerical testing were compared with known data. The comparison showed that the proposed algorithm has similar accuracy with other methods. A comparison of the speed of the proposed algorithm and a similar one based on the McCormack method is given. The conclusion is made about the superiority in speed over the McCormack method with the same accuracy.
About the Authors
Igor Ivanovich POTAPOV
Computing Center of the Far Eastern Branch of the Russian Academy of Sciences
Russian Federation
Russian Federation
Doctor of Physics and Mathematics, Professor, Head of the Department of Computational mechanics
Pavel Sergeevich TIMOSH
Computing Center of the Far Eastern Branch of the Russian Academy of Sciences
Russian Federation
Russian Federation
Research Assistant
References
1. Lax P.D. Weak Solutions of Nonlinear Hyperbolic Equations and their Numerical Computation. Communications on Pure and Applied Mathematics, vol. 7, 1954, pp. 159-193.
2. Lax P.D., Wendroff B. Systems of Conservation Laws. Communications on Pure and Applied Mathematics, vol. 13, 1960, pp. 217-237.
3. MacCormack R.W. The Effect of Viscosity in Hypervelocity Impact Cratering. AIAA Paper 69-354, 1969.
4. Von Neumann J., Richtmyer R.D. A Method for the Numerical Calculation of Hydrodynamic Shocks. Journal of Applied Physics, vol. 21, issue 3, 1950, pp. 232-237.
5. Lapidus A. A Detached Shock Calculation by Second-Order Finite Differences. Journal of Computational Physics, vol. 2, issue 2, 1967, pp. 154-177.
6. Годунов С.К. Разностный метод численного расчета разрывных решений уравнений гидродинамики. Математический сборник. Новая серия, том 47(89), вып. 3, 1959 г., стр. 271-306 / Godunov S.K. Difference method for the numerical computation of discontinuous solutions of fluid dynamics equations. Matematicheskii Sbornik. Novaya Seriya, vol. 47(89), issue 3, 1959, pp. 271-306 (in Russian).
7. Boris J.P., Book D.L. Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm that Works, J. Comp. Physics, 11, p. 38, 1973.
8. Zalesak S.T. Fully multidimensional flux-corrected transport algorithms for fluids. Journal of Computational Physics, vol. 31, issue 3, 1979, pp. 335-362.
9. Harten A. High Resolution Schemes for Hyperbolic Conservation Laws. Journal of Computational Physics, vol. 49, issue 3, 1983, pp. 357-393.
10. Chakravarthy R., Osher S. A New Class of High Resolution TVD Schemes for Hyperbolic Conservation Laws. AIAA Paper 85-0363, 1985.
11. Потапов И.И., Тимош П.С. Об использовании Центрально-разностной схемы для решения задачи газовой динамики. Информатика и системы управления, вып. 2(68), 2021 г., стр. 17‒22 / Potapov I.I., Timosh P.S. On the use of the central difference scheme for solving the problem of gas dynamics. Information Science and Control Systems, issue 2(68), 2021, pp. 17-22 (in Russian).
12. Картвелишвили Н.А. Неустановившиеся открытые потоки. Л., Гидрометеоиздат, 1968 г., 125 стр. / Kartvelishvili N.A. Unsettled open streams. L., Gidrometeoizdat, 1968, 125 p. (in Russian).
13. Беликов В.В., Алексюк А.И. Модели мелкой воды в задачах речной гидродинамики. М., РАН, 2020 г., 346 стр. / [1]. Belikov V.V., Aleksyuk A.I. Shallow water models in problems of river hydrodynamics. M., RAS, 2020, 346 p. (in Russian).
14. Amiri S.M., Talebbeydokhti N., Baghlani A. A two-dimensional well-balanced numerical model for shallow water equations. Scientia Iranica, vol. 20, issue 1, 2013, pp. 97–107.
15. Canestrelli A., Dumbser M. et al. Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Advances in Water Resources, vol. 33, issue 3, 2010, pp. 291-303.
16. Булатов О.В., Елизарова Т Г. Регуляризованные уравнения мелкой воды и эффективный метод численного моделирования течений в неглубоких водоемах. Журнал вычислительной математики и математической физики, том 51, вып. 1, 2011 г., pp. 170‒184 / Bulatov O.V., Elizarova T.G. Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows. Computational Mathematics and Mathematical Physics, vol. 51, issue1, 2011, pp. 160–173.
Review
For citations:
POTAPOV I.I., TIMOSH P.S. Solving the shallow water problem by central differences and FCT correction. Proceedings of the Institute for System Programming of the RAS (Proceedings of ISP RAS). 2022;34(5):243-250. (In Russ.) https://doi.org/10.15514/ISPRAS-2022-34(5)-17
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