张恭庆主编的《多尺度模型的基本原理》系统介绍有关多尺度建模的基本问题，主要介绍其基本原理而非具体应用。前四章介绍有关多尺度建模的一些背景材料，包括基本的物理模型，例如，连续统力学、量子力学；还包括一些多尺度问题中常用的分析工具，例如，平均方法、齐次化方法、重正规化群法、匹配渐近法等，还介绍了运用多尺度思想的经典数值方法。接下来介绍一些更前沿的内容：多物理模型的实例，即明确使用多物理渐近的分析模型，当宏观经验模型不足时，借助微观模型，使用数值方法来获取复杂系统的宏观行为规律，使用数值方法将宏观模型和微观模型结合起来，以便更好地解决局部奇点、亏量及其他问题。最后一部分主要介绍三类具体问题：带多尺度系数的微分方程、慢动力和快动力问题以及其他特殊问题。

《数学与现代科学技术丛书》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 incompressible flows

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 enserables

4.2.3 The elastic continuum limit the 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 Difficulties with the three approaches

 6.5 Seamless coupling

 6.6 Application to fluids

 6.7 Stability, accuracy and efficiency

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 Coefficients

 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

 10.5 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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