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

作     者：Xiaoming LI Shijun LIAO 

作者机构：School of Naval ArchitectureOcean and Civil EngineeringShanghai Jiao Tong UniversityShanghai 200240China State Key Laboratory of Ocean EngineeringShanghai 200240China Collaborative Innovative Center for Advanced Ship and Deep-Sea ExplorationShanghai 200240China Ministry-of-Education Key Laboratory of Scientific and Engineering ComputingShanghai 200240China 

出 版 物：《Applied Mathematics and Mechanics(English Edition)》 (应用数学和力学（英文版）)

年 卷 期：2018年第39卷第11期

页      面：1529-1546页

核心收录：

学科分类：07[理学] 0805[工学-材料科学与工程（可授工学、理学学位）] 0802[工学-机械工程] 070201[理学-理论物理] 0701[理学-数学] 0801[工学-力学（可授工学、理学学位）] 0702[理学-物理学] 

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

主　　题：chaos numerical noise clean numerical simulation(CNS) reliability of computation 

摘      要：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 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.