A new strategy to gain "clean" reliable numerical simulations of chaos and turbulence, namely the Clean Numerical Simulation (CNS), which can greatly reduce numerical noises to a tiny level much smaller than that of true solutions so numerical noises are negligible, and the corresponding numerical simulation is "clean" and thus reliable.
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.
1 Introduction.
2. CNS algorithms for temporal chaos.
3. CNS algorithms
for spatio-temporal chaos.
4. On the origin of macroscopic randomness.
5.
Ultra-chaos: a higher disorder than normal-chaos.
6. Numerical simulation of
turbulence: true or false?
7. Periodic orbits of the three-body problem.
Shijun Liao is Chun-Shen Distinguished Professor, Director, State Key Laboratory of Ocean Engineering, and Dean, School of Naval Architecture, Ocean and Civil Engineering at Shanghai Jiao Tong University, Shanghai, China. He holds a Ph.D. from Shanghai Jiao Tong University. he is also Cheung-Kong Distinguished Professor (Ministry of Education of China). His research is well-known. Topics include Nonlinear mechanics, gravity waves, turbulence, nonlinear dynamics, chaos, applied mathematics, analytic approximation method for highly nonlinear equations, reliable numerical simulations of chaotic systems and turbulence, and computer algebra methods in nonlinear mechanics. Awards and honors include Shanghai Scientific Elite (2017), National Natural Science Award (2016), Shanghai Natural Science Award (2009), Shanghai Peony Natural Science Award (2009), Shanghai Excellent Teaching Award (2004), Thomson Reuters Highly Cited Researcher in mathematics (2014, 2015, 2016), Thomson Reuters Highly Cited Researcher in engineering (2014), Worlds Top 2% Scientists 2020 (Stanford University). He is also the author of Beyond Perturbation: Introduction to the Homotopy Analysis Method, also published by CRC Press.
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