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 convergent chaotic trajectory whose numerical noise is much lower than the true physical solution, so that one can gain accurately. Thus, CNS provides for the first time an ability to check statistics stability of chaos, leading to a completely new concept of "ultra-chaos," which has both trajectory instability and statistics instability, and thus is of a higher disorder.
Notably, it is impossible to repeat experimental results of ultra-chaos even in the statistical sense. However, the reproducibility of physical experiments forms a cornerstone for modern science. Thus, ultra-chaos reveals an incompleteness of the modern science paradigm. In addition, it also reveals statistics stability as a precondition for 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 for us to enter the "clean" numerical world of chaos and turbulence.
Preface
Preface
1 Introduction
- Motivation
- A brief description of Clean Numerical Simulation
- Some illustrative applications
- Convergent chaotic trajectory
- Evolution of micro-level physical uncertainty
- Influence of small disturbances on chaos
- Discovery of periodic orbits of three-body problem
- Outline
2. CNS algorithms for temporal chaos
- Numerical noise
- Basic principles of CNS for temporal chaos
- How to reduce background numerical noise
- Determination of critical predictable time Tc
- Balance between truncation and round-off error
- Self-adaptive CNS algorithm with variable precision
- An illustrative example: Lorenz model
- Fixed time-step
- Variable optimal time-step
- Significance of convergent chaotic trajectory
3. CNS algorithms for spatio-temporal chaos
3.1 Basic principles of CNS for spatio-temporal chaos
3.1.1 How to reduce background numerical noise
3.1.2 Determination of critical predictable time Tc
3.1.3 Self-adaptive CNS algorithm with variable precision
3.2 An illustrative example: sine-Gordon equation
3.2.1 CNS algorithm for spatiotemporal chaos
3.2.2 Determination of critical predictable time Tc
3.2.3 Self-adaptive CNS algorithm with variable precision
4. On the origin of macroscopic randomness
4.1 A chaotic three-body system
4.1.1 Mathematical model and the CNS algorithm
4.1.2 From micro-level uncertainty to macroscopic randomness
4.1.3 Self-excited escape and symmetry-breaking
4.2 Turbulent Rayleigh-Bénard convection
4.2.1 Mathematical model and the CNS algorithm
4.2.2 From thermal fluctuation to macroscopic randomness
4.3 Origin of macroscopic randomness
5. Ultra-chaos: a higher disorder than normal-chaos
5.1 A new classification: normal-chaos and ultra-chaos
5.2 Examples of ultra-chaotic systems
5.2.1 Sine-Gordon equation
5.2.2 Arnold-Beltrami-Childress (ABC) flow
5.3 Possible relationships of ultra-chaos to turbulence, Poincaré section and ergodicity
5.3.1 Ultra-chaos and turbulence
5.3.2 Ultra-chaos and Poincaré section
5.3.3 Ultra-chaos and ergodicity
5.4 Influence on paradigm of scientific research
6. Numerical simulation of turbulence: true or false?
6.1 Mathematical model of turbulent Rayleigh-Bénard convection
6.2 CNS algorithm in physical space
6.3 Comparisons between CNS and DNS trajectories
6.4 Influence of small disturbances on statistics
6.5 DNS of turbulence: true or false?
6.6 Modified fourth Clay millennium problem
7. Periodic orbits of the three-body problem
7.1 Historical review
7.1.1 Era of Newton, Euler, Lagrange and Poincaré
7.1.2 Era of the electronic computer
7.2 Discovery of new periodic orbits by means of CNS
7.3 A roadmap based on CNS and artificial intelligence
7.4 Scientific significance of discovering new periodic orbits
Bibliography
Index
Biography
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), World’s 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.
