[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85068-en":3,"doc-seo-85068-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85068,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Tracking the Boundary Between Absolute/Convective Instability Using Adjoint Equations","Determining absolute/convective instability boundaries typically involves repeated saddle searches in the complex wavenumber plane and an outer scan of physical parameters to find zero absolute growth, creating nested computations that are expensive and prone to modal-branch association errors in large non-normal eigenvalue problems. A direct continuation method is developed for neutral stationary-saddle boundaries of frequency-affine generalized eigenvalue problems. The zero-group-velocity condition is written as an adjoint solvability residual and solved jointly with direct/adjoint eigenproblems, gauge constraints, and neutral-growth constraints. The resulting one-dimensional solution manifold is tracked with scaled pseudo-arclength continuation to cross parameter folds without switching the continuation variable, matching known analytical and finite-difference results to ~10^-8 and reducing wall time by about 8.1–52.2× versus nested scanning, while enabling fold geometry and re-entrant CI–AI–CI boundaries in an Oldroyd–B free-surface film flow.","arXiv :2607 .08305v2 [physics .flu-dyn] 10 Jul 2026  \nTRACKING THE BOUNDARY BETWEEN  \nABSOLUTE/CONVECTIVE INSTABILITY USING ADJOINT  \nEQUATIONS∗  \nYUE XIAO 1 , HUI LI2,3 , AND ZIJING DING2,3,*  \nAbstract. 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 developsa 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 parametric 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 EH ∼ 10 −6 suggests an estimated cost ratio of approximately 1 .8 × 104 in favour of the direct continuation. Application to an Oldroyd–B free-surface film flow reveals genuine folds of the neutral-saddle manifold and are-entrant CI–AI–CI boundary geometry for the selected saddle family.  \nKey words. Adjoint method, Pseudo-arclength continuation, Saddle point, Spatio-temporal instability  \n1 Department of Engineering Mechanics, School of Civil Engineering, Shandong University, Jinan 250061, China  \n2 School of Civil Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China  \n3 International Research Center for Intelligent Fluid Mechanics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China  \n4 Corresponding author: [z.ding@hit.edu.cn](z.ding@hit.edu.cn)  \n∗ Submitted to the editors DATE.  \nFunding: Y. Xiao is supported by the National Natural Science Foundation of China (No. 12402301) . Z. Ding is supported by the National Natural Science Foundation of China (grant nos 12102109, 52176065) and the National Key R&D programme of China (grant no. 2022YFF0503501)  \n2 Y. XIAO, H. LI, AND Z. DING  \n1. Introduction. The response of an open flow to a localised disturbance depends not only on its temporal amplification but also on the rate at which the associated wave packet is transported away from the source region. The distinction between convective and absolute instability is therefore formulated through the long-time impulse response and the analytic structure of the dispersion relation in the complex wavenumber–frequency plane [6, 7, 2, 10] . In the classical Briggs–Bers framework, an absolute mode is associated with a branch-point singularity at which spatial branches of opposite spatial causality pinch, whereas a convectively unstable disturbance is amplified but swept away from its source region. With the normal-mode convention (1.1) q′(x, y, t) = ˆq(y)exp[i(kx − ωt)],  \na neutral absolute/convective instability (AI/CI) boundary is characterised algebraically by  \ndω  \n(1.2) = 0 Im(ω) = 0  \ndk , ,  \nprovided that the stationary point is the relevant Briggs–Bers pinch [6, 16] . Thus, locating an algebraic stationary point and verifying the spatial pinch topology are closely related but logically distinct numerical tasks.  \nFor low-dimensional analytical dispersion relations, the saddle conditions may sometimes be i","cbCaia4QbArGJMYJ","https://ap.wps.com/l/cbCaia4QbArGJMYJ","pdf",633542,1,28,"English","en",105,"# Introduction\n## Absolute vs. convective instability and neutral boundaries\n## Classical Briggs–Bers framework and numerical saddle searching\n## Limitations of nested searches in multiparameter spaces\n## Prior auxiliary-equation and adjoint-based approaches","[{\"question\":\"Why do traditional methods for finding absolute/convective instability boundaries become costly?\",\"answer\":\"They typically require repeated saddle searches in the complex-wavenumber plane plus scans of physical parameters to locate the zero absolute growth condition, making the computations nested and expensive.\"},{\"question\":\"How does the proposed method replace nested saddle and parameter scans?\",\"answer\":\"It formulates the neutral stationary-saddle boundary as a direct continuation problem, expressing the zero-group-velocity condition via an adjoint solvability residual and solving it together with direct/adjoint eigenproblems and neutral-growth constraints.\"},{\"question\":\"What advantages does pseudo-arclength continuation provide in the combined parameter space?\",\"answer\":\"It tracks the resulting one-dimensional solution manifold and can cross parameter folds without switching the physical continuation variable, improving robustness while maintaining 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do traditional methods for finding absolute/convective instability boundaries become costly?","Question",{"text":75,"@type":76},"They typically require repeated saddle searches in the complex-wavenumber plane plus scans of physical parameters to locate the zero absolute growth condition, making the computations nested and expensive.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method replace nested saddle and parameter scans?",{"text":80,"@type":76},"It formulates the neutral stationary-saddle boundary as a direct continuation problem, expressing the zero-group-velocity condition via an adjoint solvability residual and solving it together with direct/adjoint eigenproblems and neutral-growth constraints.",{"name":82,"@type":73,"acceptedAnswer":83},"What advantages does pseudo-arclength continuation provide in the combined parameter space?",{"text":84,"@type":76},"It tracks the resulting one-dimensional solution manifold and can cross parameter folds 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