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.