Clean Numerical Simulation

So, CNS provides us, for the first time, with an ability to check statistics stability of chaos, which leads to a completely new concept "ultra-chaos" that has both trajectory instability and statistics instability and thus is of a higher disorder. Notably it is impossible to repeat experimental results of an ultra-chaos even in the statistical senses. However, the reproducibility of physical experiments forms a cornerstone of modern science. So, ultra-chaos reveals an incompleteness of modern science paradigm. Besides, it also reveals that statistics stability is a precondition for the use of conventional algorithms, including direct numerical simulation (DNS). In this book, several conjectures and open problems are proposed, including a modified fourth Clay millennium problem. Indeed, CNS opens a door for us to enter the "clean" numerical world of chaos and turbulence.