Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians
- Univ. of Michigan, Ann Arbor, MI (United States)
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 {open_quotes}good{close_quotes} eigenvalues are scattered, so our first point is: It is important to plot the {open_quotes}drift{close_quotes} 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 {open_quotes}anti-Kelvin{close_quotes} 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. 22 refs., 6 figs., 3 tabs.
- OSTI ID:
- 440753
- Journal Information:
- Journal of Computational Physics, Vol. 126, Issue 1; Other Information: PBD: Jun 1996
- Country of Publication:
- United States
- Language:
- English
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