Normurodov, C. B., Toyirov, A. X., & Yuldashev, S. M.
Numerical modeling of nonlinear wave systems by the spectral-grid method. |
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Full Article: PDF
Scientific Object Identifier: http://s-o-i.org/1.1/TAS-03-83-11
DOI: https://dx.doi.org/10.15863/TAS.2020.03.83.11
Language: Russian
Citation: Normurodov, C. B., Toyirov, A. X., & Yuldashev, S. M. (2020). Numerical modeling of nonlinear wave systems by the spectral-grid method. ISJ Theoretical & Applied Science, 03 (83), 43-54. Soi: http://s-o-i.org/1.1/TAS-03-83-11 Doi: https://dx.doi.org/10.15863/TAS.2020.03.83.11 |
Pages: 43-54
Published: 30.03.2020
Abstract: Numerical methods are increasingly used for the mathematical modeling of nonlinear wave systems. At the same time, their application to the solution of evolutionary problems with large gradients, described by non-stationary partial differential equations, is subject to serious difficulties. They are associated mainly with the presence of a small parameter with the oldest derivative and, as a consequence, the appearance in the solution of regions of strong spatial inhomogeneity. Therefore, the requirements imposed on the approximation property of numerical methods increase sharply. To solve these systems, spectral methods were mainly used. In this paper, the spectral-grid method is used to numerically simulate nonlinear wave systems. In the spectral-grid method, the interval of integration over the spatial variable is divided into a grid, in the grid elements the approximate solution is approximated with the help of a linear combination of a different number of series in Chebyshev polynomials of the first kind. Among the orthogonal polynomials, only Chebyshev polynomials have a minimax property, ie for these polynomials the maximum deviation from the required solution is minimal. In addition, for computational application of Chebyshev polynomials there are convenient recurrence formulas. With the help of these formulas it is easy to calculate the values of polynomials and their derivatives of the required order. When applying the spectral-grid method, the internal nodes of the introduced grid are subject to the continuity requirements of the approximate solution and its derivatives up to -th order, where is the order of the highest derivative of the differential equation. As a result of approximation of the basic differential equation, initial-boundary conditions and continuity conditions by a spectral-grid method, a system of algebraic equations is obtained. The spectral - grid method is applied to numerical modeling of initial - boundary value problems for heat conduction equations and nonlinear evolution equations. The numerical calculations performed show the high combining efficiency of the spectral-grid method.
Key words: mathematical modeling, nonlinear wave systems, spectral - grid method, evolution problems, interval of integration, approximation grid, Chebyshev polynomials of the first kind, algebraic system, approximate solution, efficiency, numerical results.
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