Tracking the boundary between absolute/convective instability using adjoint equations

Abstract

Determining absolute/convective instability boundaries conventionally requires repeated saddle searches in the complex-wavenumber plane and a subsequent scan of the physical parameter space to locate zero absolute growth. Such nested calculations become costly and sensitive to modal branch association for large non-normal eigenvalue problems. This work develops a direct continuation method for neutral stationary-saddle boundaries of frequency-affine generalised eigenvalue problems. The zero-group-velocity condition is expressed as an adjoint solvability residual and solved together with the direct and adjoint eigenproblems, complex gauge constraints and the neutral-growth condition. The resulting one-dimensional solution manifold in the combined state--parameter space is tracked by scaled pseudo-arclength continuation, allowing parameter folds to be crossed without switching the physical continuation variable. The formulation recovers the analytical Ginzburg--Landau boundary and, for a Gaussian-wake Orr--Sommerfeld problem, agrees with separately formulated finite-difference saddle corrections to approximately $10^{-8}$ in relative critical Reynolds number. Compared with nested complex-wavenumber and parameter-plane saddle scanning, the tested scans require $8.1$--$52.2$ times the wall time of the direct adjoint continuation. Extrapolation of the measured cost--accuracy trend to a boundary error of $E_H\sim10^{-6}$ suggests an estimated cost ratio of approximately $1.8\times10^{4}$ in favour of the direct continuation. Application to a coupled Oldroyd--B free-surface film reveals genuine folds of the neutral-saddle manifold and a re-entrant CI--AI--CI boundary geometry for the selected saddle family.