Gegenbauer tau methods with and without spurious eigenvalues

TL;DR

This paper proves that certain Gegenbauer tau methods accurately produce real, negative, and distinct eigenvalues for a 4th order differential eigenvalue problem, clarifying when spurious eigenvalues occur.

Contribution

It establishes rigorous conditions under which Gegenbauer tau methods yield correct eigenvalues and shows the equivalence of the modified tau approach to the Galerkin method.

Findings

Gegenbauer tau methods produce real, negative, and distinct eigenvalues for the problem.

Positive or complex eigenvalues occur outside the specified class of methods.

Modified tau approach is equivalent to the Galerkin method.

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.