NDLI: An Instability Phenomenon in Spectral Methods

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Trummer, Manfred R. Trefethen, Lloyd N.
Copyright Year	1987
Abstract	The eigenvalues of Chebyshev and Legendre spectral differentiation matrices, which determine the allowable time step in an explicit time integration, are extraordinarily sensitive to rounding errors and other perturbations. On a grid of N points per space dimension, machine rounding leads to errors in the eigenvalues of size $O(N^2 )$. This phenomenon may lead to inconsistency between predicted and observed time step restrictions. One consequence of it is that spectral differentiation by interpolation in Legendre points, which has a favorable $O(N^{ - 1} )$ time step restriction for the model problem $u_t = u_x $ in theory, is subject to an $O(N^{ - 2} )$ restriction in practice. The same effect occurs with Chebyshev points for the model problem $u_t = - xu_x $. Another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision. We know of no other examples in numerical computation of this kind of precision-dependent stability.
Starting Page	1008
Ending Page	1023
Page Count	16
File Format	PDF
ISSN	00361429
DOI	10.1137/0724066
e-ISSN	10957170
Issue Number	5
Volume Number	24
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-14
Access Restriction	Subscribed
Subject Keyword	spectral method boundary condition rounding error differentiation matrix instability Eigenvalues, eigenvectors Numerical differentiation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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