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Spectral Galerkin discretization for hydrodynamic stability problems. (English) Zbl 0967.76074
Summary: We present a spectral Galerkin discretization for calculating the eigenvalues of Orr-Sommerfeld equation. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is performed which indicates that a) no spurious eigenvalues occur, and b) reliable results can only be expected under the assumption of scale resolution, i.e., that \(\text{Re}/p^2\) is small; here Re is the Reynolds number and \(p\) is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary in order to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
76E05 Parallel shear flows in hydrodynamic stability
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
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