《数学与现代科学技术丛书6：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，同时，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

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