Article,
Computational aspects of the spectral Galerkin FEM for the Orr--Sommerfeld equation
International Journal for Numerical Methods in Fluids, 32 (1): 105--121 (Jan 15, 2000)
DOI: 10.1002/(sici)1097-0363(20000115)32:1\%3C105::aid-fld938\%3E3.0.co;2-x
Abstract
Since Orszag's paper 'Accurate solution of the Orr-Sommerfeld stability equation', J. Fluid Mech., 50, 689-703 (1971), most of the subsequent spectral techniques for solving the Orr-Sommerfeld equation (OSE) employed the Tau discretization and Chebyshev polynomials. The use of the Tau discretization appears to be accompanied by so-called spurious eigenvalues not related to the OSE and a singular matrix B in the generalized eigenvalue problem. Starting from a variational formulation of the OSE, a spectral discretization is performed using a Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions can be satisfied exactly for any spectral order and the non-singular matrices A and B are obtained in A x = lambda B x. For plane Poiseuille flow, the stiffness and the mass matrices are sparse with bandwidths 7 and 5 respectively, and the entries can be calculated explicitly (thus avoiding quadrature errors) for any polynomial flow profile U. According to the convergence results Hancke, 'Calculating large spectra in hydrodynamic stability: a p FEM approach to solve the Orr-Sommerfeld equation', Diploma Thesis, Swiss Federal Institute of Technology Zürich, Seminar for Applied Mathematics, 1998; Hancke, Melenk and Schwab, 'A spectral Galerkin method for hydrodynamic stability problems', Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zürich, no spurious eigenvalue has been found. Numerical experiments with spectral orders up to p=600 illustrate the results.
