Tracking the boundary between absolute/convective instability using adjoint equations

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 scanning calculations require $14.0$--$30.6$ times the wall time of the direct adjoint continuation, with the cost increasing as the reconstructed boundary is refined. 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. The results show that adjoint-augmented pseudo-arclength continuation can replace nested saddle searches and parameter-plane reconstruction by direct and computationally efficient tracking of the neutral boundary itself.