2022/04/13 by Jack D. Tyler, Jack Tyler, Tyler, Jack +2
Computer Science · Engineering · Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #Numerical Analysis (math.NA) #Numerical methods for differential equations #cs.NA #math.NA #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.2204.06236
Updates preprint to incorporate reviewer feedback. Main argument and results unchanged. Submitted to Journal of Computational Sciences. LaTeX, 45 pages
openalex publication_date 2022/04/13 · arxiv created 2022/07/22 · arxiv updated 2022/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In dynamical systems, it is advantageous to identify regions of flow which can exhibit maximal influence on nearby behaviour. Hyperbolic Lagrangian Coherent Structures have been introduced to obtain two-dimensional surfaces which maximise repulsion or attraction in three-dimensional dynamical systems with arbitrary time-dependence. However, the numerical method to compute them requires obtaining derivatives associated with the system, often performed through the approximation of divided differences, which can lead to significant numerical error and numerical noise. In this paper, we introduce a novel method for the numerical calculation of hyperbolic Lagrangian Coherent Structures using Differential Algebra called DA-LCS. As a form of automatic forward differentiation, it allows direct computation of the Taylor expansion of the flow, its derivatives, and the eigenvectors of the associated strain tensor, with all derivatives obtained algebraically and to machine precision. It does so without a priori information about the system, such as variational equations or explicit derivatives. We demonstrate that this can provide significant improvements in the accuracy of the Lagrangian Coherent Structures identified compared to finite-differencing methods in a series of test cases drawn from the literature. We also show how DA-LCS uncovers additional dynamical behaviour in a real-world example drawn from astrodynamics.