The influence of numerical noises on statistics computation of chaotic dynamic systems

TL;DR

This paper investigates how numerical noises affect statistical calculations in chaotic systems, revealing that such noises are negligible at equilibrium but significantly impact non-equilibrium systems, challenging the reliability of direct simulations.

Contribution

It demonstrates that numerical noises can be negligible in equilibrium but significantly distort statistics in non-equilibrium chaotic systems, highlighting limitations of current simulation methods.

Findings

Numerical noises are negligible for time-independent statistics.

Numerical noises significantly affect time-dependent statistics.

Direct numerical simulations may not reliably compute statistics for non-equilibrium systems.

Abstract

It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependance 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, and therefore numerical simulations of chaos are only mixtures of "true" solution with physical meanings and numerical noises without physical meanings. However, it is traditionally believed that statistic computations based on such kind of "mixtures" of numerical simulations of chaotic dynamic systems are acceptable. In this paper, using the so-called "clean numerical simulation" (CNS) whose numerical noises might be much smaller even than micro-level physical uncertainty and thus are negligible, we gain accurate prediction of a chaotic dynamic system in a long enough interval of time. Then, based on these reliable simulations, the influence of numerical noises on statistic computations is investigated. It is found that the influence of numerical noises is negligible when statistic results are time-independent. Unfortunately, when a chaotic dynamic system is far from equilibrium state so that its statistics are time-dependent, numerical noises have a great influence even on statistic computations. It suggests that even the direct numerical simulations (DNS) might not give reliable statistic computations for non-equilibrium dynamic systems with the SDIC.