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, and whose numerical noise is much lower than the true physical solution P so that one can gain P accurately. Thus, CNS provides for the first time an ability to check the statistics stability of chaos leading to a completely new concept of “ultra-chaos” that has both trajectory instability and statistics instability, and thus is of a higher disorder.

Notably, it is impossible to repeat the experimental results of ultra-chaos even in statistical senses. However, the reproducibility of physical experiments forms a cornerstone of modern science. Thus, ultra-chaos reveals an incompleteness of the modern scientific paradigm. In addition, it reveals that statistics stability is a precondition for the 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 to a “clean” numerical world of chaos and turbulence.