Geodesic
Convergence of linearized sequence tasks to the nonlinear sediment transport task solution
Matematičeskoe modelirovanie, Tome 29 (2017) no. 11, pp. 19-39.

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Present paper has been devoted to convergence investigation of solution of $\mathrm{2D}$ linearized bottom sediment transportation task sequence to the solution of appropriate nonlinear task in the Hilbert space $L_1$, when the value of time step tends to $0$. These models are taking into account: wave influence, bottom relief, density and porosity sediment deposits and tangential tension near the bottom for coastal zone. In previous author investigations the existence and uniqueness of $\mathrm{2D}$ linearized bottom sediment transportation task sequence have been proved and the estimation in Hilbert space $L_1$ has been done for appropriate boundary value problem in dependence of integral expressions for right side function, boundary and initial conditions. In previous author papers have been obtained results for constructed finite difference scheme convergence and stability, approximated the sequence of linearized bottom sediment transportation task. Earlier numerical results for model and real tasks have been presented for coastal zone. In given work have been presented elaboration results for convergence of linearized boundary value problem sequence solution to the solution of appropriate nonlinear task, when time step approaches to $0$. The fact of convergence of linear difference scheme solution to the solution of appropriate nonlinear bottom sediment transportation task is proved, when time and space steps tend to zero in according of results of given and previous papers.
Keywords: $\mathrm{2D}$ bottom deposits transportation model, coastal zone, nonlinear task, linearized task
Mots-clés : convergence solution investigation. 
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A. I. Sukhinov; V. V. Sidoryakina. Convergence of linearized sequence tasks to the nonlinear sediment transport task solution. Matematičeskoe modelirovanie, Tome 29 (2017) no. 11, pp. 19-39. http://geodesic.mathdoc.fr/item/MM_2017_29_11_a1/

[1] I.O. Leontev, Pribrezhnaia dinamika: volny, techeniia potoki nanosov, GEOS, M., 2001, 272 pp.

[2] G.I. Marchuk, V.P. Dymnikov, V.B. Zalesnyi, Matematicheskie modeli v geofizicheskoi gidrodinamike i chislennye metody ikh realizatsii, Gidrometeoizdat, L., 1987, 296 pp.

[3] A.D. Polianin, V.F. Zaitsev, A.I. Zhurov, Metody resheniia nelineinykh uravnenii matematicheskoi fiziki i mekhaniki, Fizmatlit, M., 2005, 256 pp.

[4] Xiaoying Liu, Shi Qi, Yuan Huang, Yuehong Chen, Pengfei Du, “Predictive modeling in sediment transportation across multiple spatial scales in the Jialing River Basin of China”, International Journal of Sediment Research, 30:3 (2015), 250–255 | DOI

[5] J.G. Venditti, M.A. Church., S.J. Bennett, “Bed form initiation from a flat sand bed”, Journal of Geophysical Research: Earth Surface, 110:1 (2005), 19 pp.

[6] P.G. Petrov, “Dvizhenie sypuchei sredy v pridonnom sloe zhidkosti”, Prikladnaia mekhanika i tekhnicheskaya fizika, 32:5 (1991), 72–75

[7] L.N. Sanne, Modelling of sand dunes in steady and tidal flow, Ph.D. Thesis, Technical University of Copenhagen, Denmark, 2003

[8] V.V. Belikov, N.M. Borisova, G.L. Gladkov, “Matematicheskaia model transporta nanosov dlia rascheta zanosimosti dnouglubitelnykh prorezei i ruslovykh karerov”, Zhurnal universiteta vodnykh kommunikacii, 2 (2010), 105–113

[9] A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko, “Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs”, Math. Mod. Comp. Simul., 6:4 (2014), 351–363 | DOI | MR | Zbl

[10] A.I. Sukhinov, V.V. Sidoriakina, “O edinstvennosti resheniia linearizovannoi dvumernoi nachalnokraevoi zadachi transporta nanosov”, Vestnik Taganrogskogo instituta imeni A. P. Chechova, 2016, no. 2, 270–274

[11] A.I. Sukhinov, A.E. Chistiakov, “Parallelnaia realizatsiia trekhmernoi modeli gidrodinamiki melkovodnykh vodoemov na supervychislitelnoi sisteme”, Vych. met. i progr., 13:1 (2012), 290–297 | Zbl

[12] A.I. Sukhinov, A.E. Chistiakov, A.A. Semeniakina, A.V. Nikitina, “Parallelnaia realizatsiia zadach transporta veshchestv i vosstanovleniia donnoi poverkhnosti na osnove skhem povyshennogo poriadka tochnosti”, Vych. met. programmirovanie, 16:2 (2015), 256–267

[13] A.I. Sukhinov, A.E. Chistyakov, E.F. Timofeeva, A.V. Shishenya, “Mathematical model for calculating coastal wave processes”, Math. Mod. Comp. Simul., 5:2 (2013), 122–129 | DOI | MR | Zbl

[14] E. Alekseenko, B. Roux, A. Sukhinov, R. Kotarba, D. Fougere, “Nonlinear hydrodynamics in a mediterranean lagoon”, Nonlinear Processes in Geophysics, 20 (2013), 189–198 | DOI

[15] E. Alekseenko, A. Sukhinov, B. Roux, R. Kotarba, D. Fougere, “Coastal hydrodynamics in a windy lagoon”, Computers Fluids, 77 (2013), 24–35 | DOI

[16] V.S. Vladimirov, Uravneniia matematicheskoi fiziki, Uchebnik, 4-e izd., ispr. i dop., Nauka, M., 1981, 512 pp.

[17] A.N. Tikhonov, A.A. Samarskii, Uravneniia matematicheskoi fiziki, Nauka, M., 1977, 735 pp.