Multi-scale phenomena in complex fluids: modeling, analysis and numerical simulation : [collection of lecture notes generated from the first two series of mini-courses of "Shanghai Summer School on Analysis and Numerics in Modern Sciences" held during the

Contents Preface Zhaojun Bai, Wenbin Chen, Richard Scalettar, Ichitaro Yamazaki: Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model Abstract 1 Hubbard model and QMC simulations 1.1 Hubbard model 1.1.1 Hubbard model with no hopping 1.1.2 Hubbard model without interaction 1.2 Determinant QMC 1.2.1 Computable approximation of distribution operator P 1.2.2 Algorithm 1.2.3 Physical measurements 1.3 Hybrid QMC 1.3.1 Computable approximation of distribution operator P 1.3.2 Algorithm 1.3.3 Physical measurements 2 Hubbard matrix analysis 2.1 Hubbard matrix 2.2 Basic properties 2.3 Matrix exponential B = et rK 2.4 Eigenvalue distribution of M 2.4.1 The case U = 0 2.4.2 The case U = 0 2.5 Condition number of M 2.5.1 The case U = 0 2.5.2 The case U = 0 2.6 Condition number of M(k) 3 Self-adaptive direct linear solvers 3.1 Block cyclic reduction 3.2 Block structural orthogonal factorization 3.3 A hybrid method 3.4 Self-adaptive reduction factor k 3.5 Self-adaptive block cyclic reduction method 3.6 Numerical experiments 4 Preconditioned iterative linear solvers 4.1 Iterative solvers and preconditioning 4.2 Previous work 4.3 Cholesky factorization 4.4 Incomplete Cholesky factorizations 4.4.1 IC 4.4.2 Modified IC 4.5 Robust incomplete Cholesky preconditioners 4.5.1 RICl 4.5.2 RIC2 4.5.3 RIC3 4.6 Performance evaluation 4.6.1 Moderately interacting systems 4.6.2 Strongly interacting systems Appendix A. Updating algorithm in DQMC A.1 Rank-one updates A.2 Metropolis ratio and Green's function computations Appendix B Particle-hole transformation B.1 Algebraic identities B.2 Particle-hole transformation in DQMC B.3 Particle-hole transformation in the HQMC B.4 Some identities of matrix exponentials Acknowledgments References Albert C. Fannjiang: Introduction to Propagation, Time Reversal and Imaging in Random Media 1 Scalar diffraction theory 1.1 Introduction 1.2 Kirchhoff's theory of diffraction 1.3 Huygens-Fresnel principle 1.4 Fresnel and Fraunhofer diffraction 1.5 Focal spot size and resolution 2 Approximations: weak fluctuation 2.1 Born approximation 2.2 Rytovapproximation 2.3 The extended Huygens-Fresnel principle 2.4 Paraxial approximation 3 The Wigner distribution 4 Markovian approximation 4.1 White-noise scaling 4.2 Markovian limit 5 Two-frequency transport theory 5.1 Paraxial waves 5.1.1 Two-frequency radiative transfer equations 5.1.2 The longitudinal and transverse cases 5.2 Spherical waves 5.2.1 Geometrical radiative transfer 5.2.2 Spatial (frequency) spread and coherence bandwidth 5.2.3 Small-scale asymptotics 6 Application: time reversal 6.1 Spherical wave 6.2 Paraxial wave 6.3 Anomalous focal spot 6.4 Duality and turbulence-induced aperture 6.5 Coherence length 6.6 Broadband time reversal communications 7 Application: imaging in random media 7.1 Imaging of phase objects 7.2 Long-exposure imaging 7.3 Short-exposure imaging 7.4 Coherent imaging of multiple point targets in Rician media 7.4.1 Differential scattered field in clutter 7.4.2 Imaging functions 7.4.3 Numerical simulation with a Rician medium 7.5 Coherent imaging in a Rayleigh medium Acknowledgement References Thomas Y. Hou: Multiscale Computations for Flow and Transport in Porous Media Abstract 1 Introduction 2 Review of homogenization theory 2.1 Homogenization theory for elliptic problems 2.2 Homogenization for hyperbolic problems 2.3 Convection of microstructure 3 Numerical homogenization based on sampling techniques 3.1 Convergence of the particle method 3.2 Vortex methods for incompressible flows 4 Numerical upscaling based on multiscale finite element methods 4.1 Multiscale finite element methods for elliptic PDEs 4.2 Error estimates (h ) 4.4 The over-sampling technique 4.5 Performance and implementation issues 4.6 Applications 4.7 Brief overview of mixed finite element and finite volume element methods 4.8 MsFEM using limited global information 4.9 Analysis 5 Multiscale finite element methods for nonlinear partial differential equations 5.1 Multiscale finite volume element method (MsFVEM) 5.2 Examples of V h 10 5.3 Convergence of MsFEM for nonlinear partial differential equations 5.4 Multiscale finite element methods for nonlinear parabolic equations 5.5 Numerical results 5.6 Generalizations of MsFEM and some remarks 6 Multiscale simulations of two-phase immiscible flow in adaptive coordinate system 6.1 Numerical averaging across streamlines 6.2 N urnerical results 7 Conclusions References Chun Liu: An Introduction of Elastic Complex Fluids: An Energetic Variational Approach Abstract 1 Introduction 2 Calculus of variations 2.1 Euler-Lagrange equations 2.2 Direct methods 2.3 Convexity 2.4 Dynamics 2.5 Hamilton's principle 2.5.1 Flow map and deformation tensor 2.5.2 Variation of the domain v.s. variation of the function 2.5.3 Least action principle 2.6 Constraint problems 2.6.1 Harmonic maps 2.6.2 Liquid crystals 2.6.3 Methods of penalty 3 Navier-Stokes equation 3.1 Newtonian fluids 3.1.1 Existence of global weak solution 3.1.2 Existence of classical solution 3.1.3 Regularity 3.1.4 Partial regularity 4 Viscoelastic materials 4.1 Flow map and deformation tensor 4.2 Force balance and Oldroyd-B systems 4.3 Energetic variational formulation 5 Liquid crystal flows 5.1 Ericksen-Leslie theory 5.2 Existence and regularity 6 Free interface motion in mixtures 6.1 An energetic variational approach with phase field method 6.2 Marangoni-Benard convection 6.3 Mixtures involving liquid crystals 7 Magneto hydrodynamics (MHD) 7.1 Introduction 7.2 The evolution of the magnetic field 7.3 The energy law 7.4 The linear momentum equation 7.5 The dynamics of magnetic field lines References Qi Wang: Introduction to Kinetic Theory for Complex Fluids Abstract 1 Introduction 2 A primer for equilibrium thermodynamics 3 Basics of statistical mechanics 4 Equilibrium distribution of the end-toend vector in simple polymer models 5 Kinetic theory for polymers 5.1 Langevin equation 5.2 System of constraints 5.3 Bead-spring (Rouse) chain model References Multi-Scale Phenomena in Complex Fluids is a collection of lecture notes delivered during the first two series of mini-courses from “Shanghai Summer School on Analysis and Numerics in Modern Sciences”, which was held in 2004 and 2006 at Fudan University, Shanghai, China.This review volume of 5 chapters, covering various fields in complex fluids, places emphasis on multi-scale modeling, analyses and simulations. It will be of special interest to researchers and graduate students who want to work in the field of complex fluids. Multi-Scale Phenomena in Complex Fluids is a collection of lecture notes delivered during the first two series of mini-courses from Shanghai Summer School on Analysis and Numerics in Modern Sciences , which was held in 2004 and 2006 at Fudan University, Shanghai, China. This review volume of 5 chapters, covering various fields in complex fluids, places emphasis on multi-scale modeling, analyses and simulations. It will be of special interest to researchers and graduate students who want to work in the field of complex fluids. Multi-scale Phenomena In Complex Fluids Is A Collection Of Lecture Notes Delivered During The arst Two Series Of Mini-courses From Shanghai Summer School On Analysis And Numerics In Modern Sciences, Which Was Held In 2004 And 2006 At Fudan University, Shanghai, China. This Review Volume Of 5 Chapters, Covering Various Fields In Complex Fluids, Places Emphasis On Multi-scale Modeling, Analyses And Simulations. It Will Be Of Special Interest To Researchers And Graduate Students Who Want To Work In The Field Of Complex Fluids. A collection of lecture notes delivered during the two series of mini-courses from Shanghai Summer School on Analysis and Numerics in Modern Sciences, which was held in 2004 and 2006 at Fudan University, Shanghai, China. Covering various fields in complex fluids, it places emphasis on multi-scale modeling, analyses and simulations.