Clean Numerical Simulation

Preface

Preface

1. Motivation
2. A brief description of Clean Numerical Simulation
3. Some illustrative applications

1. Convergent chaotic trajectory
2. Evolution of micro-level physical uncertainty
3. Influence of small disturbances on chaos
4. Discovery of periodic orbits of three-body problem

4. Outline

1. Numerical noise
2. Basic principles of CNS for temporal chaos

1. How to reduce background numerical noise
2. Determination of critical predictable time Tc
3. Balance between truncation and round-off error
4. Self-adaptive CNS algorithm with variable precision

3. An illustrative example: Lorenz model

1. Fixed time-step
2. Variable optimal time-step

4. Significance of convergent chaotic trajectory

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

4.1 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

5.1 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

6.1 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

          7.1 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

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.