LAX-STABILITY OF FULLY DISCRETE SPECTRAL METHODS VIA STABILITY REGIONS AND PSEUDO-EIGENVALUES

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep 'method of lines' approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the ε-pseudo-eigenvalues of the spatial discretization operator lie within O(ε) of the stability region as ε → 0. An ε-pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with ∥E ∥ ≤ ε; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of ut = -xux on [-1, 1] is Lax-stable if and only if the time step satisfies k = O(N-2), although eigenvalue analysis would suggest a much weaker restriction of the form k ≤ CN-1. © 1990.