Arbitrary precision computation of hydrodynamic stability eigenvalues

TL;DR

This work tackles the challenge of accurately computing spectra for non-normal matrices arising in hydrodynamic stability, where conventional double precision can give misleading eigenvalues. It introduces an arbitrary-precision Chebyshev tau-QZ framework, with two discretizations of the Orr-Sommerfeld equation: a D2 formulation and a D4 formulation, to compute eigenvalues $c$ for Poiseuille and Couette flows at high $Re$. The study demonstrates that high working precision is essential to resolve far-spectrum modes, illustrating convergence via Hausdorff distances and highlighting trade-offs between matrix size and numerical conditioning. The findings show that while D2 and D4 can yield accurate spectra, high precision can enable faster computation with D4, making high-precision spectral analysis feasible for challenging non-normal problems with broad spectral content.

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.

Figures (18)

• Figure 1: The Hausdorff distances of the spectra $\sigma_P(A^{7\times 7}_{-1})$ (left) and $\sigma_P(A^{7\times 7}_{1})$ (right) to the correct spectra $\sigma(A^{7\times 7}_{-1})$ (left) and $\sigma(A^{7\times 7}_{1})$ (right) for different numbers of significant bits $P$ or respective machine precisions $\varepsilon_P$. The distance decreases like $\mathcal{O}(\varepsilon_P^{1/2})$ in the case $s = 1$ and like $\mathcal{O}\left(\varepsilon_P\right)$ in the case $s = -1$. The dashed red and blue lines indicate the machine precision $\varepsilon_{\texttt{double}} = 2.2204\cdot 10^{-16}$ for standard double precision with 53 significant bits and $\varepsilon_{\texttt{ext}} = 1.9259\cdot 10^{-34}$ for extended precision with 113 significant bits, respectively.
• Figure 2: The spectra $\sigma_P(A^{7\times 7}_{1})$ (left) and $\sigma_P(A^{7\times 7}_{-1})$ (right) that we compute for different numbers of significant bits $P$. For high numbers of $P$, we clearly obtain the correct spectra given in \ref{['eq:spectrum_Ass_m1', 'eq:spectrum_Ass_p1']}.
• Figure 3: The spectrum for plane Poiseuille flow with $\mathrm{Re} = 10^4$, $a = 1$, computed in double precision with $N=200$ polynomials for the even- and odd eigenfunctions. This reproduces the result in Fig. 1 of Straughan1996.
• Figure 4: Poiseuille flow, $\mathrm{Re} = 10^{5}$. The distances as described in \ref{['eq:def_distances']} for Poiseuille flow with $Re = 10^{5}$ and different values of $P$. We note that that increasing the number of polynomials does not lead to a decrease of the error if the number of significant bits is not chosen high enough.
• Figure 5: Poiseuille flow, $\mathrm{Re} = 2\cdot10^{5}$. The distances as described in \ref{['eq:def_distances']} for Poiseuille flow with $Re = 2\cdot10^5$ and different values of $P$. We note that that increasing the number of polynomials does not lead to a decrease of the error if the number of significant bits is not chosen high enough.