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Fast Construction of Forward Flow Maps using Eulerian Based Interpolation Schemes

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Abstract

We propose a modification to a recently developed Eulerian interpolation scheme for constructing the flow map for autonomous, periodic and aperiodic dynamical systems. We show that the proposed methods significantly improve the computational efficiency when a large number of flow maps are needed, yet retain second-order accuracy as in the original approach. The idea is to pre-compute and to store a carefully selected set of intermediate flow maps. When the initial and the final time of a required flow map are known, our proposed methods can simply load these pre-processed short-time flow maps for flow map construction. Numerical examples are included to validate our theoretical prediction, and demonstrate the effectiveness of these proposed Eulerian interpolation schemes as a simple numerical tool for other applications such as the finite-time Lyapunov exponent in determining the Lagrangian coherent structure of dynamical systems and the geometrical optics problems.

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Acknowledgements

The work of You was supported in part by the Natural Science Foundation of Jiangsu Province (BK20171071, BK20161053) and the National Natural Science Foundation of China (11701287). The second author’s work was supported by the Hong Kong RGC under Grant 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, Hong Kong, China

    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. Fast Construction of Forward Flow Maps using Eulerian Based Interpolation Schemes. J Sci Comput 82, 32 (2020). https://doi.org/10.1007/s10915-020-01141-z

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  • DOI: https://doi.org/10.1007/s10915-020-01141-z

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