An Instability Phenomenon in Spectral Methods

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.