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A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

Part of: Basic methods in fluid mechanics Ergodic theory Approximation methods and numerical treatment of dynamical systems

Published online by Cambridge University Press:  03 July 2015

Guoqiao You  and
Shingyu Leung
Guoqiao You
Affiliation:
The School of Science, Nanjing Audit University, Nanjing, Jiangsu Province, China
Shingyu Leung*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
*
*Corresponding author. Email addresses: youguoqiao@sina.com (G. You), masyleung@ust.hk (S. Leung)
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Abstract

We propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

Keywords

Numerical methods for PDEslevel set methoddynamical systemsflow visualization

MSC classification

Secondary: 37A25: Ergodicity, mixing, rates of mixing 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) 76M27: Visualization algorithms
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 203 - 229
Copyright
Copyright © Global-Science Press 2015 

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