Clean Numerical Simulation: A New Strategy to Obtain Reliable Solutions of Chaotic Dynamic Systems

TL;DR

The paper introduces the Clean Numerical Simulation (CNS) strategy, which significantly reduces numerical noise to reliably simulate chaotic systems over long periods, overcoming limitations of traditional methods.

Contribution

It presents the CNS approach as a novel method for obtaining accurate long-term solutions of chaotic systems by minimizing numerical errors.

Findings

CNS provides convergent, reliable trajectories for Hamiltonian chaotic systems.

Traditional double-precision methods fail to produce accurate Fourier spectra and autocorrelation functions.

Statistical properties of chaotic systems are often misrepresented by conventional numerical algorithms.

Abstract

It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, 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, together with its applications 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 such as three-body problem, and thus should have many applications in various fields. We find that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions. In addition, it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. Thus, our CNS results strongly suggest that one had better to be very careful on DNS results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when turbulent flows are in a statistical stationary state.