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Article information
2023 , Volume 28, ą 3, p.41-59
Fedotova Z.I., Khakimzyanov G.S.
Nonlinear-dispersive models of wave hydrodynamics: hierarchy of models, high accuracy, numerical algorithms
The article presents the results of a study of nonlinear dispersion (NLD) models and numerical methods for their implementation from the standpoint of a hierarchical approach, which allows formulating chains of “shallow water” models, the derivation of which starts from the fully nonlinear equations with dispersion and moves towards simplified models with inheritance of the most important properties. This study addresses the Serre – Green – Naghdi (SGN) model of the second long-wavelength approximation O(µ2), the modification of which allows obtaining a new mSGN model with improved phase characteristics due to the increased fourth order accuracy that approximates the dispersion relation of the “reference” model of potential fluid flows with a free boundary. A similar modification for the model of the fourth long-wavelength approximation O(µ4) leads to the increase in the accuracy of the dispersion relation up to the eighth order. Note that the order of approximations is understood to be the exponent of the long-wavelength parameter O(µn), where µ = d/λ, d and λ are the average values of the depth of the water area and the wavelength, respectively. The paper shows that the modified models describe the characteristics of the “reference” model in the case of a horizontal bottom more accurately than the SGN model, as along with the change in phase velocity, amplitude, length and steepness of surface waves as they move from the deep part of the water area to the shallow water in the case of uneven bottom. Previous approaches to construct and study numerical algorithms for solving SGN equations can also be used for models of increased accuracy. In this paper, we demonstrated that a predictorcorrector difference scheme approximating a one-dimensional mSGN model realizes this possibility. It is shown that the stability condition for this scheme allows using Courant numbers greater than one, while for a similar difference scheme in the case of dispersionless shallow water equations, the Courant number is less than one. The stability condition for the mSGN scheme is more restrictive than the condition for the scheme approximating the SGN equations, but the mSGN scheme is preferable in terms of the suppression of shortwavelength harmonics. It is also shown that there are values of the mSGN scheme parameters for which the dispersion of the difference scheme does not suppress the dispersion of the mSGNb model
Keywords: long surface waves, uneven bottom, nonlinear dispersion equations, hierarchy of models, dispersion relation, phase velocity, Green law, analysis of dissipation and dispersion of difference schemes
doi: 10.25743/ICT.2023.28.3.004
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: 3144-0877 References:
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Bibliography link:
Fedotova Z.I., Khakimzyanov G.S. Nonlinear-dispersive models of wave hydrodynamics: hierarchy of models, high accuracy, numerical algorithms // Computational technologies. 2023. V. 28. ą 3. P. 41-59
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