%0 Journal Article %1 melenk2000spectral %A Melenk, J. M. %A Kirchner, N. P. %A Schwab, C. %D 2000 %J Computing %K 76e05-parallel-shear-flows 76m22-spectral-methods-in-fluid-mechanics %N 2 %P 97--118 %R 10.1007/s006070070014 %T Spectral Galerkin Discretization for Hydrodynamic Stability Problems %U https://doi.org/10.1007/s006070070014 %V 65 %X A spectral Galerkin discretization 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 scale resolution, i.e., that Re/p2 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.

@article{melenk2000spectral, abstract = {A spectral Galerkin discretization 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 scale resolution, i.e., that Re/p2 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.}, added-at = {2020-07-09T07:56:22.000+0200}, author = {Melenk, J. M. and Kirchner, N. P. and Schwab, C.}, biburl = {https://www.bibsonomy.org/bibtex/259f22c8884f448fb98f7065a3eeabfe2/gdmcbain}, day = 01, doi = {10.1007/s006070070014}, interhash = {a47060cb82561c4fa2b82c116e1b4d4b}, intrahash = {59f22c8884f448fb98f7065a3eeabfe2}, issn = {1436-5057}, journal = {Computing}, keywords = {76e05-parallel-shear-flows 76m22-spectral-methods-in-fluid-mechanics}, month = nov, number = 2, pages = {97--118}, timestamp = {2020-07-09T07:56:22.000+0200}, title = {Spectral {G}alerkin Discretization for Hydrodynamic Stability Problems}, url = {https://doi.org/10.1007/s006070070014}, volume = 65, year = 2000 }