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Cybernetics and programming

The increase in the rate of convergence of finite difference method based on the use of middleware solutions

Korobeinikov Anatolii Grigor'evich

Doctor of Technical Science

professor, Pushkov institute of terrestrial magnetism, ionosphere and radio wave propagation of the Russian Academy of Sciences St.-Petersburg Filial

199034, Russia, g. Saint Petersburg, ul. Mendeleevskaya, 1

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Grishentsev Aleksei Yur'evich

Doctor of Technical Science

Associate Professor, St. Petersburg National Research University of Information Technologies, Mechanics and Optics

197101, Russia, St. Petersburg, Kronverkskiy prospect, d. 49

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Abstract: To study the characteristics of the process of functioning of any system of mathematical methods, including the engine, should be carried out formalization of the process, ie. a mathematical model. By mathematical modeling we mean the process of establishing compliance with this real object of a mathematical object (mathematical model), and the study of the mathematical model, which allows to obtain characteristics of this real object. A type of a mathematical model depends both on the nature of the real object and tasks of the research and the required reliability and accuracy of the solution of this problem. Any mathematical model, like any other, describes the real object only with a certain degree of approximation to reality. This paper presents a method for calculating interim solution in n-dimensional problem with boundary conditions, contributing to the acceleration of the convergence process of a finite difference method. In the practical implementation of this method the number of iterations to achieve a given residual was reduced to 10 - 100 times, due to the search of the intermediate solutions. Thus, this method can be used to significantly improve the efficiency of a finite difference method.

Keywords: numerical methods, finite elements, finite differences, FEM, MKP, differential equations, mathematical model, stability, n-dimensional problem, mathematical modeling
This article written in Russian. You can find full text of article in Russian here .

Grishentsev A. Yu., Korobeinikov A. G. Razrabotka modeli resheniya obratnoi zadachi vertikal'nogo zondirovaniya ionosfery// Nauchno-tekhnicheskii vest nik SPb GU ITMO-SPb: SPbGU ITMO, 2011, 2(72)-s.109-113.
Demidovich B. P., Maron I. A., Shuvalova E. Z. Chislennye metody analiza. Priblizhenie funktsii, differentsial'nye i integral'nye uravneniya. – SPb.: Lan', 2010. – 400 s.
Bakhvalov N.S., Voevodin V.V. Sovremennye problemy vy-chislitel'noi matematiki i matematicheskogo modelirovaniya: v 2 t., T. 1. / In-t vychislitel'noi matematiki. – M.: Nauka, 2005, 343 s.
Formalev V. F., Reveznikov D. L. Chislennye metody. – M.: Fizmatlit, 2004. – 400 s.
Korobeinikov A. G., Kudrin P. A., Sidorkina I. G. Algoritm raspoznavaniya trekhmernykh izobrazhenii s vysokoi detalizatsiei. Vestnik Mariiskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Radiotekhnicheskie i infokommunikatsionnye sistemy, 2010 №2, s. 91-98.
Samarskii A.A., Gulin A.V. Chislennye metody: Ucheb. Posobie dlya vuzov.–M.: Nauka. Gl. red. fiz.–mat. lit., 1989. – 432s.
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