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Volume 31, Issue 5
Eulerian Algorithms for Computing the Forward Finite Time Lyapunov Exponent Without Finite Difference upon the Flow Map

Guoqiao You, Changfeng Xue & Shaozhong Deng

Commun. Comput. Phys., 31 (2022), pp. 1467-1488.

Published online: 2022-05

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  • Abstract

In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The advantages of the proposed algorithms mainly manifest in two aspects. First, previous Eulerian approaches for computing the FTLE field are improved so that the Jacobian of the flow map can be obtained by directly solving a corresponding system of partial differential equations, rather than by implementing certain finite difference upon the flow map, which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges. Second, in the proposed algorithms, all computations are done on the fly, that is, all required partial differential equations are solved forward in time, which is practically more natural. The new algorithms still maintain the optimal computational complexity as well as the second order accuracy. Numerical examples demonstrate the effectiveness of the proposed algorithms.

  • AMS Subject Headings

37A25, 37M25, 76M27

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COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1467, author = {You , GuoqiaoXue , Changfeng and Deng , Shaozhong}, title = {Eulerian Algorithms for Computing the Forward Finite Time Lyapunov Exponent Without Finite Difference upon the Flow Map}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {1467--1488}, abstract = {

In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The advantages of the proposed algorithms mainly manifest in two aspects. First, previous Eulerian approaches for computing the FTLE field are improved so that the Jacobian of the flow map can be obtained by directly solving a corresponding system of partial differential equations, rather than by implementing certain finite difference upon the flow map, which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges. Second, in the proposed algorithms, all computations are done on the fly, that is, all required partial differential equations are solved forward in time, which is practically more natural. The new algorithms still maintain the optimal computational complexity as well as the second order accuracy. Numerical examples demonstrate the effectiveness of the proposed algorithms.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0193}, url = {http://global-sci.org/intro/article_detail/cicp/20511.html} }
TY - JOUR T1 - Eulerian Algorithms for Computing the Forward Finite Time Lyapunov Exponent Without Finite Difference upon the Flow Map AU - You , Guoqiao AU - Xue , Changfeng AU - Deng , Shaozhong JO - Communications in Computational Physics VL - 5 SP - 1467 EP - 1488 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0193 UR - https://global-sci.org/intro/article_detail/cicp/20511.html KW - Finite time Lyapunov exponent, flow map, flow visualization, dynamical system. AB -

In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The advantages of the proposed algorithms mainly manifest in two aspects. First, previous Eulerian approaches for computing the FTLE field are improved so that the Jacobian of the flow map can be obtained by directly solving a corresponding system of partial differential equations, rather than by implementing certain finite difference upon the flow map, which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges. Second, in the proposed algorithms, all computations are done on the fly, that is, all required partial differential equations are solved forward in time, which is practically more natural. The new algorithms still maintain the optimal computational complexity as well as the second order accuracy. Numerical examples demonstrate the effectiveness of the proposed algorithms.

Guoqiao You, Changfeng Xue & Shaozhong Deng. (2022). Eulerian Algorithms for Computing the Forward Finite Time Lyapunov Exponent Without Finite Difference upon the Flow Map. Communications in Computational Physics. 31 (5). 1467-1488. doi:10.4208/cicp.OA-2021-0193
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