Convergence and roundoff errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

We show that an efficient way for solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method is fast and preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two sources of error and investigate the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to roundoff errors, even when apparently good spectral convergence is achieved. The influence of roundoff errors is analyzed using the spectral portrait technique and by comparison between double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and roundoff errors on eigenvalues and eigenvectors.