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An Improved Eulerian Approach for the Finite Time Lyapunov Exponent

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Abstract

We propose a new Eulerian numerical approach to compute the Jacobian of flow maps in continuous dynamical systems and subsequently the so-called finite time Lyapunov exponent (FTLE) for Lagrangian coherent structure extraction. The original approach computes the flow map and then numerically determines the Jacobian of the map using finite differences. The new algorithm improves the original Eulerian formulation so that we first obtain partial differential equations for each component of the Jacobian and then solve these equations to obtain the required Jacobian. For periodic dynamical systems, based on the time doubling technique developed for computing the longtime flow map, we also propose a new efficient iterative method to compute the Jacobian of the longtime flow map. Numerical examples will demonstrate that our new proposed approach is more accurate than the original one in computing the Jacobian and thus the FTLE field, especially near the FTLE ridges.

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Acknowledgements

The work of You was supported by the National Natural Science Foundation of China (No. 11701287) and the Natural Science Foundation of Jiangsu Province (No. BK20171071). The work of Leung was supported in part by the Hong Kong RGC Grants 16303114 and 16309316.

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Authors and Affiliations

  1. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China

    Guoqiao You

  2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Shingyu Leung

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  1. Guoqiao You
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  2. Shingyu Leung
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Correspondence to Guoqiao You.

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You, G., Leung, S. An Improved Eulerian Approach for the Finite Time Lyapunov Exponent. J Sci Comput 76, 1407–1435 (2018). https://doi.org/10.1007/s10915-018-0669-y

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  • DOI: https://doi.org/10.1007/s10915-018-0669-y

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