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Article information
2017 , Volume 22, ¹ 5, p.73-109
Fedotova Z.I., Khakimzyanov G.S., Gusev O.I., Shokina N.Y.
History of the development and analysis of numerical methods for solving nonlinear dispersive equations of hydrodynamics. II. Two-dimensional models
The review of the finite-difference methods for solving two-dimensional NLD-equations is presented. It is found that, despite of the successful application of these methods to the problems of wave hydrodynamics, the works on the theoretical investigation of the used difference schemes are nearly absent. In order to fill the gap in this knowledge, the investigation of stability and dispersion properties is done for the series of difference schemes. Although the simple mathematical tools were used for the analysis, the detailed computations are presented in order to make future investigations of the properties of similar finite-difference schemes easier. One of the main conclusions of this work is that the stability conditions for the difference schemes of the 2D-equations with dispersion are weaker than the similar conditions for the schemes which approximate the nondispersive shallow water equations. Therefore, this property, which was earlier discovered by the authors for one-dimensional schemes, is inherited by the two-dimensional schemes. The form of the stability conditions for the two-dimensional schemes also proved to be similar to the one-dimensional case. Nevertheless, such inheritance is not true for the dispersion property. In 1D-case, the value of the Courant number typically exists, for which the phase error of the scheme becomes minimal. But for the discussed two-dimensional the schemes similar property was not discovered. The conducted investigations set a number of new problems. In particular, it becomes clear that it is necessary to develop the schemes, which are invariant with respect to rotation in order to improve the description of the dispersion properties of the model, in particular, to minimize the dependency of dispersion on the direction of the wave vector.
[full text]
Keywords: nonlinear dispersive equations, two-dimensional models, numerical algorithms, finite-difference methods, stability, dispersion
Author(s):
Fedotova Zinaida Ivanovna
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Lavrentiev ave. 6
Phone Office: (383) 334-91-21
E-mail: zf@ict.nsc.ru
Khakimzyanov Gayaz Salimovich
Dr. , Professor
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: khak@ict.nsc.ru
SPIN-code: 3144-0877Gusev Oleg Igorevitch
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, 6, Acad. Lavrentjev avenue
Phone Office: (383) 334-91-18
E-mail: GusevOI@ict.sbras.ru
SPIN-code: 3995-2134Shokina Nina Yurievna
PhD.
Position: Research Scientist
Office: Medical Center University of Freiburg
Address: 79106, Germany, Freiburg, Killianstrasse, 5a
Phone Office: (49761) 270 73930
E-mail: nina.shokina@uniklinik-freiburg.de
SPIN-code: 8680-7439 References:
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Bibliography link:
Fedotova Z.I., Khakimzyanov G.S., Gusev O.I., Shokina N.Y. History of the development and analysis of numerical methods for solving nonlinear dispersive equations of hydrodynamics. II. Two-dimensional models // Computational technologies. 2017. V. 22. ¹ 5. P. 73-109
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