The work objective is to study the four-layer scheme convergence rate. The problem of finding an approximate solution to the linear operator equation Au = f is considered. Two-layer and three-layer iterative methods are used to solve this problem. At that, the three-layer conjugate directions methods converge faster than the two-layer gradient methods. The research problem is to establish whether the four-layer scheme has a speed advantage as compared to the three-layer scheme. The four-layer scheme is constructed, and its parameters are calculated for this purpose. It is proved that the four-layer iterative scheme of a variational type for solving finite-difference equations downs to the three-layer scheme.
finite-difference equations, three-layer scheme, four-layer scheme, variational methods.
Большинство прикладных задач таких, как задача транспорта веществ [1–3], гидродинамики мелководных водоемов [4–5], аэродинамики [6–7], динамики популяций [8] и других, сводятся к решению системы линейных алгебраических уравнений (СЛАУ). Для решения таких систем уравнений используются двух- и трехслойные итерационные схемы.
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2. Sukhinov, A.I., et al. Parallel´naya realizatsiya zadach transporta veshchestv i vosstanovleniya donnoy poverkhnosti na osnove skhem povyshennogo poryadka tochnosti. [Parallel implementation of transport tasks substances and restore the bottom surface on the basis of high order schemes.] Parallel´nye vychislitel´nye tekhnologii (PaVT´2015). Trudy mezhdunarodnoy nauchnoy konferentsii. [Parallel Computing Technologies (PaVT’2015). Proc. Int. Sci. Conf.] 2015, pp. 285-296 (in Russian).
3. Sukhinov, A.I., Chistyakov, A.E., Protsenko, E.A. Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs. Mathematical Models and Computer Simulations, 2014, vol. 6, no. 4, pp. 351-363.
4. Sukhinov, A.I., Chistyakov, A.E., Alekseenko, E.V. Chislennaya realizatsiya trekhmernoy modeli gidrodinamiki dlya melkovodnykh vodoemov na supervychislitel´noy sisteme. [Numerical realization of three-dimensional hydrodynamic model for shallow water basins on supercomputing system.] Mathematical Models and Computer Simulations, 2011, vol. 23, no. 3, pp. 3-21 (in Russian).
5. Sukhinov, A.I., Belova, Y.V. Matematicheskaya model´ transformatsii form fosfora, azota i kremniya v dvizhushcheysya turbulentnoy vodnoy srede v zadachakh dinamiki planktonnykh populyatsiy. [Mathematical model of phosphorus, nitrogen and silicon forms transformation in moving turbulent water environment in problems of plankton population dynamics.] Engineering Journal of Don, 2015, vol. 37, no. 3, pp. 50 (in Russian).
6. Sukhinov, A.I., Khachunts, D.S., Chistyakov, A.E. A mathematical model of pollutant propagation in near-ground atmospheric layer of a coastal region and its software implementation. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 7, pp. 1216-1231.
7. Sukhinov, A.I., Khachunts, D.S., Chistyakov, A.E. Matematicheskaya model´ rasprostraneniya primesi v prizemnom sloe atmosfery i ee programmnaya realizatsiya na mnogoprotsessornoy vychislitel´noy sisteme. [Mathematical model of impurities in the atmospheric boundary layer and its program implementation on a multiprocessor computer system.] Vestnik UGATU, 2015, vol. 19, no. 1, pp. 185-195 (in Russian).
8. Sukhinov, A.I., Nikitina, A.V., Chistyakov, A.E. Modelirovanie stsenariya biologicheskoy reabilitatsii azovskogo morya. [Numerical simulation of biological remediation Azov Sea.] Mathematical Models and Computer Simulations, 2012, vol. 24, no. 9, pp. 3-21 (in Russian).
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