Determine the physical status of discretized eigenmodes in LAADS

Determine how many eigenmodes the continuous linear perturbation equations admit and establish a reliable method for distinguishing physical modes from discretization artifacts, including the possibility of false-positive and false-negative classifications in LAADS.

Background

LAADS solves a discretized version of the linear perturbation equations governing reflex instabilities in accretion discs. The number of eigenmodes returned by the numerical calculation depends directly on the discretization resolution, whereas the number of modes admitted by the underlying continuous equations is not known.

The authors report that separating genuine physical modes from numerical artifacts is particularly difficult when exploring the high-dimensional parameter space of reflex instabilities. LAADS applies sanity checks to remove many artificial modes, but the authors cannot guarantee that all classifications are correct.

References

We do not know how many eigenmodes the continuous linear perturbation equations admit, but usually a lot less. Distinguishing between discretisation artefacts and physical modes is a hard problem which, to the best of our knowledge, has not been solved yet. LAADS runs a few sanity checks to eliminate the bulk of the artificial modes (red dots), but we cannot guarantee that there is no false negative or false positive.

— Reflex instabilities II: The reflex motion of the central object drives at least seven instabilities in accretion discs  (2610.08472 - Magnan et al., 6 Oct 2026) in Figure 1 caption, Exploration strategy

In fact, we could not find convincing evidence for pressurised bracket modes in the stellocentric reference frame. We used a continuation algorithm to find the pressurised eigenvalue that connects to the pressure-less eigenvalue, but Fig.~\ref{fig:Bracket_modes}-top left shows that the corresponding pressurised eigenvector has suspicious features such as rapid oscillations and jumps, and no sign of convergence with resolution. This suggests that this pressurised eigenmode is a numerical artefact induced by the discretisation of the eigenvalue problem formed by Eqs.~eq:Fourier-transformed_equations~and~eq:Fourier_transformed_equations_indirect_potential. We therefore conjecture that the stellocentric reference frame does not allow for pressurised bracket modes.

— Reflex instabilities II: The reflex motion of the central object drives at least seven instabilities in accretion discs  (2610.08472 - Magnan et al., 6 Oct 2026) in Section 5, “Bracket modes”