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Title: Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians

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 {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 truemore » 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.« less

Authors:
 [1]
  1. Univ. of Michigan, Ann Arbor, MI (United States)
Publication Date:
OSTI Identifier:
440753
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 126; Journal Issue: 1; Other Information: PBD: Jun 1996
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; TIDE; LAPLACE EQUATION; EIGENVALUES; SEAS; BOUNDARY CONDITIONS

Citation Formats

Boyd, J P. Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians. United States: N. p., 1996. Web. doi:10.1006/jcph.1996.0116.
Boyd, J P. Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians. United States. https://doi.org/10.1006/jcph.1996.0116
Boyd, J P. 1996. "Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians". United States. https://doi.org/10.1006/jcph.1996.0116.
@article{osti_440753,
title = {Traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians},
author = {Boyd, J P},
abstractNote = {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.},
doi = {10.1006/jcph.1996.0116},
url = {https://www.osti.gov/biblio/440753}, journal = {Journal of Computational Physics},
number = 1,
volume = 126,
place = {United States},
year = {Sat Jun 01 00:00:00 EDT 1996},
month = {Sat Jun 01 00:00:00 EDT 1996}
}