%0 Journal Article %1 charalambides2009gegenbauer %A Charalambides, Marios %A Waleffe, Fabian %D 2009 %I Society for Industrial & Applied Mathematics (SIAM) %J SIAM Journal on Numerical Analysis %K 26c10-polynomials-location-of-zeros 65d30-numerical-integration 65l10-numerical-analysis-odes-bvps 65l15-odes-eigenvalue-problems 65m70-pdes-spectral-collocation-and-related-methods 65n35-pdes-bvps-spectral-collocation-and-related-methods %N 1 %P 48--68 %R 10.1137/070704228 %T Gegenbauer Tau Methods With and Without Spurious Eigenvalues %U https://doi.org/10.1137%2F070704228 %V 47 %X It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative and distinct eigenvalues, as is the case for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and `inviscid' Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results show that the results are sharp: positive or complex eigenvalues arise outside of this class. The widely used modified tau approach is proved to be equivalent to the Galerkin method.

@article{charalambides2009gegenbauer, abstract = { It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative and distinct eigenvalues, as is the case for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and `inviscid' Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results show that the results are sharp: positive or complex eigenvalues arise outside of this class. The widely used modified tau approach is proved to be equivalent to the Galerkin method. }, added-at = {2019-11-24T23:19:25.000+0100}, author = {Charalambides, Marios and Waleffe, Fabian}, biburl = {https://www.bibsonomy.org/bibtex/23889537e35395e45405b056b8ee846a7/gdmcbain}, doi = {10.1137/070704228}, interhash = {ab3f72edf8d28eedac009876668fb488}, intrahash = {3889537e35395e45405b056b8ee846a7}, journal = {SIAM Journal on Numerical Analysis}, keywords = {26c10-polynomials-location-of-zeros 65d30-numerical-integration 65l10-numerical-analysis-odes-bvps 65l15-odes-eigenvalue-problems 65m70-pdes-spectral-collocation-and-related-methods 65n35-pdes-bvps-spectral-collocation-and-related-methods}, month = jan, number = 1, pages = {48--68}, publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})}, timestamp = {2019-11-24T23:19:25.000+0100}, title = {Gegenbauer Tau Methods With and Without Spurious Eigenvalues}, url = {https://doi.org/10.1137%2F070704228}, volume = 47, year = 2009 }