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《多尺度模型的基本原理(英文版)》是由科學出版社出版的。
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《數學與現代科學技術叢書》序
Preface
Chapter 1 Introduction
1.1 Examples of multiscale problems
1.1.1 Multiscale data and their representation
1.1.2 Differential equations with multiscale data
1.1.3 Differential equations with small parameters
1.2 Multi-physics problems
1.2.1 Examples of scale-dependent phenomena
1.2.2 Deficiencies of the traditional approaches to modeling
1.2.3 The multi-physics modeling hierarchy
1.3 Analytical methods
1.4 Numerical methods
1.4.1 Linear scaling algorithms
1.4.2 Sublinear scaling algorithms
1.4.3 Type A and type B multiscale problems
1.4.4 Concurrent vs. sequential coupling
1.5 What are the main challenges?
1.6 Notes
Bibliography
Chapter 2 Analytical Methods
2.1 Matched asymptotics
2.1.1 A simple advection-diffusion equation
2.1.2 Boundary layers in incompressibleflows
2.1.3 Structure and dynamics of shocks
2.1.4 Transition layers in the Allen-Cahn equation
2.2 The WKB method
2.3 Averaging methods
2.3.1 Oscillatory problems
2.3.2 Stochastic ordinary differential equations
2.3.3 Stochastic simulation algorithms
2.4 Multiscale expansions
2.4.1 Removing secular terms
2.4.2 Homogenization of elliptic equations
2.4.3 Homogenization of the Hamilton-Jacobi equations
2.4.4 Flow in porous media
2.5 Scaling and self-similar solutions
2.5.1 Dimensional analysis
2.5.2 Self-similar solutions of PDEs
2.6 Renormalization group analysis
2.6.1 The Ising model and critical exponents
2.6.2 An illustration of the renormalization transformation
2.6.3 RG analysis of the two-dimensional Ising model
2.6.4 A PDE example
2.7 The Mori-Zwanzig formalism
2.8 Notes
Bibliography
Chapter 3 Classical Multiscale Algorithms
3.1 Multigrid method
3.2 Fast summation methods
3.2.1 Low rank kernels
3.2.2 Hierarchical algorithms
3.2.3 The fast multi-pole method
3.3 Adaptive mesh refinement
3.3.1 A posteriori error estimates and local error indicators
3.3.2 The moving mesh method
3.4 Domain decomposition methods
3.4.1 Non-overlapping domain decomposition methods
3.4.2 Overlapping domain decomposition methods
3.5 Multiscale representation
3.5.1 Hierarchical bases
3.5.2 Multi-resolution analysis and wavelet bases
3.6 Notes
Bibliography
Chapter 4 The Hierarchy of Physical Models
4.1 Continuum mechanics
4.1.1 Stress and strain in solids
4.1.2 Variational principles in elasticity theory
4.1.3 Conservation laws
4.1.4 Dynamic theory of solids and thermoelasticity
4.1.5 Dynamics of fluids
4.2 Molecular dynamics
4.2.1 Empirical potentials
4.2.2 Equilibrium states and ensembles
4.2.3 The elastic continuum limitthe Cauchy-Born rule
4.2.4 Non-equilibrium theory
4.2.5 Linear response theory and the Green-Kubo formula
4.3 Kinetic theory
4.3.1 The BBGKY hierarchy
4.3.2 The Boltzmann equation
4.3.3 The equilibrium states
4.3.4 Macroscopic conservation laws
4.3.5 The hydrodynamic regime
4.3.6 Other kinetic models
4.4 Electronic structure models
4.4.1 The quantum many-body problem
4.4.2 Hartree and Hartree-Fock approximation
4.4.3 Density functional theory
4.4.4 Tight-binding models
4.5 Notes
Bibliography
Chapter 5 Examples of Multi-physics Models
5.1 Brownian dynamics models of polymer fluids
5.2 Extensions of the Cauchy-Born rule
5.2.1 High order, exponential and local Cauchy-Born rules
5.2.2 An example of a one-dimensional chain
5.2.3 Sheets and nanotubes
5.3 The moving contact line problem
5.3.1 Classical continuum theory
5.3.2 Improved continuum models
5.3.3 Measuring the boundary conditions using molecular dynamics
5.4 Notes
Bibliography
Chapter 6 Capturing the Macroscale Behavior
6.1 Some classical examples
6.1.1 The Car-Parrinello molecular dynamics
6.1.2 The quasi-continuum method
6.1.3 The kinetic scheme
6.1.4 Cloud-resolving convection parametrization
6.2 Multi-grid and the equation-free approach
6.2.1 Extended multi-grid method
6.2.2 The equation-free approach
6.3 The heterogeneous multiscale method
6.3.1 The main components of HMM
6.3.2 Simulating gas dynamics using molecular dynamics
6.3.3 The classical examples from the HMM viewpoint
6.3.4 Modifying traditional algorithms to handle multiscale problems
6.4 Some general remarks
6.4.1 Similarities and differences
6.4.2 Diffculties with the three approaches
6.5 Seamless coupling
6.6 Application to fluids
6.7 Stability, accuracy and effciency
6.7.1 The heterogeneous multiscale method
6.7.2 The boosting algorithm
6.7.3 The equation-free approach
6.8 Notes
Bibliography
Chapter 7 Resolving Local Events or Singularities
7.1 Domain decomposition method
7.1.1 Energy-based formulation
7.1.2 Dynamic atomistic and continuum methods for solids
7.1.3 Coupled atomistic and continuum methods for fluids
7.2 Adaptive model refinement or model reduction
7.2.1 The nonlocal quasicontinuum method
7.2.2 Coupled gas dynamic-kinetic models
7.3 The heterogeneous multiscale method
7.4 Stability issues
7.5 Consistency issues illustrated using QC
7.5.1 The appearance of the ghost force
7.5.2 Removing the ghost force
7.5.3 Truncation error analysis
7.6 Notes
Bibliography
Chapter 8 Elliptic Equations with Multiscale Coeffcients
8.1 Multiscale finite element methods
8.1.1 The generalized finite element method
8.1.2 Residual-free bubbles
8.1.3 Variational multiscale methods
8.1.4 Multiscale basis functions
8.1.5 Relations between the various methods
8.2 Upscaling via successive elimination of small scale components
8.3 Sublinear scaling algorithms
8.3.1 Finite element HMM
8.3.2 The local microscale problem
8.3.3 Error estimates
8.3.4 Information about the gradients
8.4 Notes
Bibliography
Chapter 9 Problems with Multiple Time Scales
9.1 ODEs with disparate time scales
9.1.1 General setup for limit theorems
9.1.2 Implicit methods
9.1.3 Stablized Runge-Kutta methods
9.1.4 HMM
9.2 Application of HMM to stochastic simulation algorithms
9.3 Coarse-grained molecular dynamics
9.4 Notes
Bibliography
Chapter 10 Rare Events
10.1 Theoretical background
10.1.1 Metastable states and reduction to Markov chains
10.1.2 Transition state theory
10.1.3 Large deviation theory
10.1.4 First exit times
10.1.5 Transition path theory
10.2 Numerical algorithms
10.2.1 Finding transition states
10.2.2 Finding the minimal energy path
10.2.3 Finding the transition path ensemble or the transition tubes Transition path sampling
10.3 Accelerated dynamics
10.3.1 TST-based acceleration techniques
10.3.2 Metadynamics
10.4 Notes
Bibliography
Chapter 11 Some Perspectives
11.1 Top-down and bottom-up
11.2 Problems without scale separation
11.2.1 Variational model reduction
11.2.2 Modeling memory effects
Bibliography
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