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Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

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Abstract

It is well-known that chaotic dynamic systems, e.g., three-body system and turbulent flow, have sensitive dependence on the initial conditions (SDIC). Unfortunately, numerical noises, i.e., truncation error and round-off error, always exist in practice. Thus, due to the SDIC, the long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e., the clean numerical simulation (CNS), is briefly described, and applied to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems, and thus should have many applications in various fields. It is found that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions (ACFs). In addition, even the statistic properties of chaotic systems cannot be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. The CNS results strongly suggest that one had better to be very careful on the direct numerical simulation (DNS) results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when the turbulent flow is in a statistical stationary state.

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Acknowledgement

We thank the anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Xiaoming Li & Shijun Liao

  2. State Key Laboratory of Ocean Engineering, Shanghai, 200240, China

    Shijun Liao

  3. Collaborative Innovative Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China

    Shijun Liao

  4. Ministry-of-Education Key Laboratory of Scientific and Engineering Computing, Shanghai, 200240, China

    Shijun Liao

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  1. Xiaoming Li
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  2. Shijun Liao
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Correspondence to Shijun Liao.

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Project supported by the National Natural Science Foundation of China (No. 91752104)

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Li, X., Liao, S. Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. Appl. Math. Mech.-Engl. Ed. 39, 1529–1546 (2018). https://doi.org/10.1007/s10483-018-2383-6

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  • DOI: https://doi.org/10.1007/s10483-018-2383-6

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