Clean Numerical Simulation

Clean Numerical Simulation

"This book proposes a new strategy to gain "clean" reliable numerical simulations of chaos and turbulence, namely the Clean Numerical Simulation (CNS), which can greatly reduce numerical noises to a tiny level much smaller than that of true solutions so that numerical noises are negligible, and the corresponding numerical simulation is "clean" and thus reliable. By means of the CNS, one can gain "clean" reliable numerical simulations of chaotic systems in a long enough interval of time. The CNS provides a much more reliable tool than ever to study chaotic and turbulent systems in various fields of science and engineering. In this book, the author will describe the basic ideas of the CNS and illustrate the CNS as a new powerful tool for chaos and turbulence. Unlike traditional algorithms in single/double precision, the CNS can provide us the "clean" results of chaos and turbulence. Using these "clean" results as benchmark solutions, we can investigate many important scientific problems, such as the origin of macroscopic randomness, the influence of micro-level uncertainty and tiny external noises to chaos and turbulence, and so on. Comparing the "clean" CNS results with those badly "polluted" by numerical noises, one can investigate the influence of database noises on machine learning. Through this application and more, the author illustrates the validity and potential of the CNS. The CNS is as a new, powerful tool for chaos and turbulence. It opens a new door for us to enter a "clean" numerical world"--

Clean Numerical Simulation

This book describes the basic principles of Clean Numerical Simulation (CNS) proposed by the author in 2009, as well as several of its applications. Unlike conventional algorithms, CNS gives in a sufficiently long interval of time a convergent chaotic trajectory whose numerical noise is much lower than the true physical solution, so that one can gain accurately. Thus, CNS provides for the first time an ability to check statistics stability of chaos, leading to a completely new concept of "ultra-chaos," which has both trajectory instability and statistics instability, and thus is of a higher disorder. Notably, it is impossible to repeat experimental results of ultra-chaos even in the statistical sense. However, the reproducibility of physical experiments forms a cornerstone for modern science. Thus, ultra-chaos reveals an incompleteness of the modern science paradigm. In addition, it also reveals statistics stability as a precondition for use of conventional algorithms, including direct numerical simulation (DNS). In Clean Numerical Simulation, several conjectures and open problems are proposed, including a modified fourth Clay millennium problem. Indeed, CNS opens the door for us to enter the "clean" numerical world of chaos and turbulence.

Direct Numerical Simulations