Arbitrary precision computation of hydrodynamic stability eigenvalues
Journal:
arXiv
Published Date:
Apr 30, 2025
Abstract
We show that by using higher order precision arithmetic, i.e., using floating
point types with more significant bits than standard double precision numbers,
one may accurately compute eigenvalues for non-normal matrices arising in
hydrodynamic stability problems. The basic principle is illustrated by a
classical example of two $7\times 7$ matrices for which it is well known that
eigenvalue computations fail when using standard double precision arithmetic.
We then present an implementation of the Chebyshev tau-QZ method allowing the
use of a large number of Chebyshev polynomials together with arbitrary
precision arithmetic. This is used to compute the behavior of the spectra for
Couette and Poiseuille flow at high Reynolds number. An experimental
convergence analysis finally makes it evident that high order precision is
required to obtain accurate results.
