Development of an algorithm for calculating stable solutions of the Saint-Venant equation using an upwind implicit difference scheme

Authors

DOI:

https://doi.org/10.15587/1729-4061.2021.239148

Keywords:

Saint-Venant equations, hyperbolic system, implicit scheme, upwind difference scheme, stability

Abstract

The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that:

1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region;

2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC;

3) the proposed computational algorithm is stable;

4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable);

5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time.

Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.

Author Biographies

Rakhmatillo Aloev, National University of Uzbekistan

Doctor of Math-Physics Sciences, Professor

Department Computational Mathematics and Information Systems

Abdumauvlen Berdyshev, Abai Kazakh National Pedagogical University; Institute of Information and Computational Technologies

Doctor of Math-Physics Sciences, Professor, Head of Department

Department of Mathematics and Mathematical Modeling

Institute of Mathematics, Physics and Informatics

Сhief Researcher

Aziza Akbarova, National University of Uzbekistan

Postgraduate Student

Department Computational Mathematics and Information Systems

Zharasbek Baishemirov, Abai Kazakh National Pedagogical University; Institute of Information and Computational Technologies

PhD, Associate Professor

Department of Mathematics and Mathematical Modeling

Institute of Mathematics, Physics and Informatics

Сhief Researcher

 

References

  1. Klimovich, V. I., Petrov, O. A. (2012). Chislennoe modelirovanie techenii pri rabote vodoslivnoi plotiny Bureiskoi GES. Izvestiia Vserossiiskogo nauchno-issledovatelskogo instituta gidrotekhniki im. B. E. Vedeneeva, 266, 22–37.
  2. Tsyganova, M. V., Lemeshko, E. M. (2017). The shelf water dynamics in the Danube delta region based on numerical simulation. Krym – ekologo-ekonomicheskii region. Prostranstvo noosfernogo razvitiia. Sevastopol, 260–262. Available at: https://www.elibrary.ru/item.asp?id=30118605
  3. Rakhuba, A. V., Shmakova, M. V. (2015). Mathematical modeling of the dynamics of sedimentation as a factor in eutrophication of the water masses of the Kuibyshev reservoir. Izvestiia Samarskogo nauchnogo tsentra Rossiiskoi akademii nauk, 17 (4), 189–193. Available at: https://cyberleninka.ru/article/n/matematicheskoe-modelirovanie-dinamiki-zaileniya-kak-faktora-evtrofirovaniya-vodnyh-mass-kuybyshevskogo-vodohranilischa
  4. Sheverdiaev, I. V., Berdnikov, S. V., Kleschenkov, A. V. (2017). HEC-RAS using for hydrologic regime modeling on the Don’s delta. Ekologiia. Ekonomika. Informatika. Seriia: Sistemnyianaliz i modelirovanie ekonomicheskikh i ekologicheskikh sistem, 1 (2), 113–122. Available at: https://www.elibrary.ru/item.asp?id=30298680
  5. Bogomolov, A. V., Lepikhin, A. P., Tiunov, A. A. (2014). Ispolzovanie chislennykh gidrodinamicheskikh modelei dlia otsenki effektivnosti proektnykh reshenii po zaschite beregov (na primere reki Don v raione goroda Pavlovska). Vodnoe khoziaistvo Rossii: problemy, tekhnologii, upravlenie, 1, 50–57. Available at: https://cyberleninka.ru/article/n/ispolzovanie-chislennyh-gidrodinamicheskih-modeley-dlya-otsenki-effektivnosti-proektnyh-resheniy-po-zaschite-beregov-na-primere
  6. Oshkin, M. I., Pisarev, A. V., Zheltobriukhov, V. F., Polozova, I. A., Kartushina, Iu. N. (2015). Ispolzovanie kompiuternogo modelirovaniia dinamiki poverkhnostnykh vod reki Medveditsy dlia resheniia prirodookhrannykh zadach. Vestnik Kazanskogo tekhnologicheskogo universiteta, 18 (18), 246–248. Available at: https://cyberleninka.ru/article/n/ispolzovanie-kompyuternogo-modelirovaniya-dinamiki-poverhnostnyh-vod-reki-medveditsy-dlya-resheniya-prirodoohrannyh-zadach
  7. Vasilev, O. F., Godunov, S. K., Pritvits, N. A., Temnoeva, T. A., Friazinova, I. L., Shugrin, S. M. (1963). Chislennii metod rascheta rasprostraneniia dlinnykh voln v otkrytykh ruslakh i prilozhenie ego k zadache o pavodke. Doklady AN SSSR, 151 (3), 525–527. Available at: http://mi.mathnet.ru/dan28337
  8. Hayat, A., Shang, P. (2019). A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope. Automatica, 100, 52–60. doi: http://doi.org/10.1016/j.automatica.2018.10.035
  9. Godunov, S. K. (1979). Uravneniia matematicheskoi fiziki. Moscow: «Nauka», 392. Available at: http://eqworld.ipmnet.ru/ru/library/books/Godunov1979ru.djvu
  10. Bastin, G., Coron, J. M. (2016). Stability and Boundary Stabilization of 1-D Hyperbolic Systems. itemirkhauser Basel. Springer International Publishing Switzerland, 307. doi: http://doi.org/10.1007/978-3-319-32062-5
  11. Göttlich, S., Schillen, P. (2017). Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization. European Journal of Control, 35, 11–18. doi: http://doi.org/10.1016/j.ejcon.2017.02.002
  12. Bastin, G., Coron, J.-M. (2017). A quadratic Lyapunov function for hyperbolic density–velocity systems with nonuniform steady states. Systems & Control Letters, 104, 66–71. doi: http://doi.org/10.1016/j.sysconle.2017.03.013
  13. Bastin, G., Coron, J.-M., d’ Andréa-Novel, B. (2009). On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks & Heterogeneous Media, 4 (2), 177–187. doi: http://doi.org/10.3934/nhm.2009.4.177
  14. Coron, J.-M., Bastin, G. (2015). Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the C1-Norm. SIAM Journal on Control and Optimization, 53 (3), 1464–1483. doi: http://doi.org/10.1137/14097080x
  15. Coron, J.-M., Hu, L., Olive, G. (2017). Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation. Automatica, 84, 95–100. doi: http://doi.org/10.1016/j.automatica.2017.05.013
  16. Peskin, C. S. (2002). The immersed boundary method. Acta Numerica, 11, 479–517. doi: http://doi.org/10.1017/s0962492902000077
  17. Volkov, K. N. (2005). Realizatsiia skhemy rasschepleniia na raznesennoi setke dlia rascheta nestatsionarnykh techenii viazkoi neszhimaemoi zhidkosti. Vychislitelnye metody i programmirovaniia, 6 (1), 269–282. Available at: http://mi.mathnet.ru/vmp648
  18. Blokhin, A. M., Aloev, R. D., Hudayberganov, M. U. (2014). One Class of Stable Difference Schemes for Hyperbolic System. American Journal of Numerical Analysis, 2 (3), 85–89. Available at: http://pubs.sciepub.com/ajna/2/3/4
  19. Aloev, R. D., Khudoyberganov, M. U., Blokhin, A. M. (2018). Construction and research of adequate computational models for quasilinear hyperbolic systems. Numerical Algebra, Control & Optimization, 8 (3), 287–299. doi: http://doi.org/10.3934/naco.2018017
  20. Berdyshev, A., Imomnazarov, K., Tang, J.-G., Mikhailov, A. (2016). The Laguerre spectral method as applied to numerical solution of a two-dimensional linear dynamic seismic problem for porous media. Open Computer Science, 6 (1), 208–212. doi: http://doi.org/10.1515/comp-2016-0018
  21. Diagne, A., Diagne, M., Tang, S., Krstic, M. (2015). Backstepping stabilization of the linearized Saint-Venant-Exner Model: Part II- output feedback. 2015 54th IEEE Conference on Decision and Control (CDC), 1248–1253. doi: http://doi.org/10.1109/cdc.2015.7402382
  22. Samarskii, A. A., Nikolaev, E. S. (1978). Metody resheniia setochnykh uravnenii. Moscow: «Nauka», 532. Available at: http://samarskii.ru/books/book1978.pdf

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Published

2021-08-30

How to Cite

Aloev, R., Berdyshev, A., Akbarova, A., & Baishemirov, Z. (2021). Development of an algorithm for calculating stable solutions of the Saint-Venant equation using an upwind implicit difference scheme . Eastern-European Journal of Enterprise Technologies, 4(4(112), 47–56. https://doi.org/10.15587/1729-4061.2021.239148

Issue

Vol. 4 No. 4(112) (2021): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2021 Rakhmatillo Aloev, Abdumauvlen Berdyshev, Aziza Akbarova, Zharasbek Baishemirov

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