Author： Boyd J.P.
Publisher： Academic Press
ISSN： 0021-9991
Source： Journal of Computational Physics, Vol.126, Iss.1, 1996-06, pp. : 11-20
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Abstract
We make several observations about eigenvalue problems using, as examples, Laplace's tidal equations and the differential equation satisfied by the associated Legendre functions. Whatever the discretization, only some of the eigenvalues of the N -dimensional matrix eigenvalue problem will be good approximations to those of the differential equation- usually the N /2 eigenvalues of smallest magnitude. For the tidal problem, however, the "good" eigenvalues are scattered, so our first point is: It is important to plot the "drift" of eigenvalues with changes in resolution. We suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue, divided by the magnitude of the eigenvalue or the intermodal separation, whichever is smaller. Second, as a final safeguard, it is important to look at the Chebyshev coefficients of the mode: We show a numerically computed "anti-Kelvin" wave which has little eigenvalue drift, but is completely spurious as is obvious from its spectral series. Third, inverting the roles of parameters can drastically modify the spectrum; Legendre's equation may have either an infinite number of discrete modes or only a handful, depending on which parameter is the eigenvalue. Fourth, when the modes are singular but decay to zero at the endpoints (as is true of tides), a tanh-mapping can retrieve the usual exponential accuracy of spectral methods. Fifth, the pseudospectral method is more reliable than deriving a banded Galerkin matrix by means of recurrence relations; the pseudospectral code is simple to check, whereas it is easy to make an untestable mistake with the intricate algebra required for the Galerkin method. Sixth, we offer a brief cautionary tale about overlooked modes. All these cautions are applicable to all forms of spatial discretization including finite difference and finite element methods. However, we limit our illustrations to spectral schemes, where these difficulties are most easily resolved. With a bit of care, the pseudospectral method is a very robust and efficient method for solving differential eigenproblems, even with singular eigenmodes.
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