Influence of numerical noises on computer-generated simulation of spatio-temporal chaos

Abstract

• An efficient strategy to gain reliable simulations of patio-temporal chaotic systems in a long interval of time. • The numerical noises have great influence on the numerical simulations of chaos, not only quantitative but also qualitatively. • Numerical simulations in a long interval of time should be checked very carefully. Due to the famous butterfly-effect, it is rather difficult to gain a reliable simulation of chaos in an interval of time that is long enough from statistic viewpoint. In this paper we propose an efficient numerical strategy of the clean numerical simulation (CNS) in physical space to gain reliable simulation of spatio-temporal chaos in a long enough interval of time by means of reducing both of truncation and round-off error to a required tiny level. Without loss of generality, the damped driven sine-Gordon equation is used to illustrate its validity for spatio-temporal chaos. The reliable result in a long interval of time given by this CNS algorithm is used as benchmark solution to investigate the influence of numerical noise (i.e. truncation and round-off error) by comparing it in details with those given by the Runge-Kutta’s method in double precision. It is found that numerical noise can lead to, both quantitatively and qualitatively, huge deviation of the spatio-temporal chaotic system not only in trajectories but also even in statistics ! Therefore, in practice we should check very carefully the reliability of computer-generated simulations of spatio-temporal chaotic systems (including turbulent flows), especially when the duration of simulation is quite long. The new strategy of the CNS in physical space provides us an efficient way to accurately study lots of spatio-temporal chaotic systems in science and engineering in a long enough interval of time so as to reveal some new physical truths by means of clearing away numerical noises.