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A Spectral Galerkin Method for Hydrodynamic Stability Problems

by N. P. Hancke and J. M. Melenk and Ch. Schwab

(Report number 1998-06)

Abstract
A spectral Galerkin method for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of {\em scale resolution}, i.e., that $\ren/p^2$ is small; here $\ren$ is the Reynolds number and p is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically.

Keywords: Orr-Sommerfeld equation, hydrodynamic stability, eigenvalue problem, spectral method

@Techreport{HMS98_231,
  author = {N. P. Hancke and J. M. Melenk and Ch. Schwab},
  title = {A Spectral Galerkin Method for Hydrodynamic Stability Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1998-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1998/1998-06.pdf },
  year = {1998}
}

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