Clean Numerical Simulation

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