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Resumen
The Chebyshev-tau method is a popular and useful method for approximating solutions to boundary value problems. For certain stability problems the method yields a set of "spurious eigenvalues". In these problems all of the eigenvalues are known to be negative, but the Chebyshev-tau methods returns at least one positive eigenvalue along with approximations to the actual eigenvalues. These eigenvalues clearly do not belong and so historically have been called spurious eigenvalues. The spurious eigenvalues can lead to a false assumption that a system is unstable when in reality it is not. In the past spurious eigenvalues were assumed to be due to discretization errors that would disappear for large enough truncation order or they were eliminated ad-hoc.
In this work a model problem that is simple enough to easily work with, yet exhibits the desired behavior is studied. Gegenbauer polynomials will be used in the tau method in order to simultaneously study a range of tau methods that include the Chebyshev- and Legendre-tau methods. It will be shown that when the Legendre-tau method is used to approximate the eigenvalues of the model problem an infinite generalized eigenvalue arises for every truncation order. Using generalized eigenvalue theory it will also be shown that for a range of Gegenbauer polynomials, including the Chebyshev polynomials, the Gegenbauer-tau method produces an approximation to this infinite generalized eigenvalue for every truncation order. This approximation will be shown to grow, in magnitude, like $N\sp4$ where N is the truncation order of the method. Hence, it will be shown that the spurious eigenvalues are not numerical errors, but in fact belong in the solution for every truncation order.
