Gegenbauer Tau Methods With and Without Spurious Eigenvalues

Abstract

It is proven that a class of Gegenbauer tau approximations to a fourth order differential eigenvalue problem of a hydrodynamic type provides 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.