Clean Numerical Simulation

This book describes the basic principles of the Clean Numerical Simulation (CNS), proposed by the author in 2009, and several of its applications. Unlike conventional algorithms, CNS can give convergent chaotic trajectory in a sufficiently long interval of time, whose numerical noise is much lower than the true physical solution so that one can gain accurately. 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.