MATHEMATICAL MODELING OF THE HYDRODYNAMIC STABILITY PROBLEM BY THE SPECTRAL-GRID METHOD

Abstract

In the article, using the method of small perturbations, mathematical models of hydrodynamic stability for single-phase flows are obtained. The spectral-grid method is used to approximate the stability equations. It combines the high accuracy of the spectral method of nonuniform grids and allows one to determine all the eigenvalues of the problem under consideration at once. In the spectral-grid method, the interval of integration with respect to the spatial variable is divided into a grid; in the grid elements, the approximate solution is approximated using a linear combination of a different number of series in Chebyshev polynomials of the first kind. Among orthogonal polynomials, only Chebyshev polynomials have the minimax property, i.e. for these polynomials, the maximum deviation from the desired solution is minimal. In addition, there are convenient recurrence formulas for the computational application of Chebyshev polynomials. Using these formulas, you can easily calculate the values of polynomials and their derivatives of the desired order