本书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。
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目录
- Contents
《数学与现代科学技术丛书》序
Preface
Chapter 1 Introduction 1
1.1 Examples of multiscale problems 1
1.1.1 Multiscale data and their representation 1
1.1.2 Differential equations with multiscale data 2
1.1.3 Differential equations with small parameters 4
1.2 Multi-physics problems 4
1.2.1 Examples of scale-dependent phenomena 4
1.2.2 Differential equations of the dependant approach to modeling 6
1.2.3 The multi-physics modeling hierarchy 9
1.3 Analytical methods 10
1.4 Numerical algorithms 11
1.4.1 Linear scaling algorithms 11
1.4.2 Sublinear scaling algorithms 12
1.4.3 Type A and type B multiscale problems 12
1.4.4 Concurrent multiscale coupling 13
1.5 What are the main challenges? 15
1.6 Notes 16
Bibliography 18
Chapter 2 Analytical Methods 22
2.1 Matched asymptotics 22
2.1.1 A simple advection-diffusion equation 23
2.1.2 Boundary layers in incompressible flows 24
2.1.3 Structure and dynamics of shocks 26
2.1.4 Transition layers in the Allen-Cahn equation 28
2.2 The WKB method 30
2.3 Averaging methods 32
2.3.1 Oscillatory problems 33
2.3.2 Stochastic ordinary differential equations 36
2.3.3 Stochastic simulation algorithms 40
2.4 Multiscale expansions 47
2.4.1 Removing secular terms 47
2.4.2 Homogenization of elliptic equations 49
2.4.3 Homogenization of the Hamilton-Jacobi equations 52
2.4.4 Flow in porous media 55
2.5 Scaling and self-similar solutions 56
2.5.1 Dimensional analysis 56
2.5.2 Self-similar solutions of PDEs 57
2.6 Renormalization group analysis 61
2.6.1 The Ising model and critical exponents 61
2.6.2 An illustration of the renormalization transformation 64
2.6.3 RG analysis of the two-dimensional Ising model 66
2.6.4 A PDE example 69
2.7 The Mori-Zwanzig formalism 71
2.8 Notes 75
Bibliography 75
Chapter 3 Classical Multiscale Algorithms 79
3.1 Multigrid method 79
3.2 Fast summation methods 87
3.2.1 Low rank kernels 88
3.2.2 Hierarchical algorithms 90
3.2.3 The fast multipole method 94
3.3 Adaptive mesh refinement 96
3.3.1 A posterior error estimates and local error indicators 97
3.3.2 The moving mesh method 99
3.4 Domain decomposition methods 100
3.4.1 Non-overlapping domain decomposition methods 101
3.4.2 Overlapping domain decomposition methods 103
3.5 Multiscale representations 104
3.5.1 Hierarchical bases 105
3.5.2 Multi-resolution analysis and wavelet bases 106
3.6 Notes 112
Bibliography 112
Chapter 4 The Hierarchy of Physical Models 115
4.1 Continuum mechanics 116
4.1.1 Stress and strain in solids 118
4.1.2 Variational principles in elasticity theory 120
4.1.3 Conservation laws 123
4.1.4 Dynamic theory of solids and thermoelasticity 125
4.1.5 Dynamics of fluids 127
4.2 Molecular dynamics 131
4.2.1 Empirical potentials 131
4.2.2 Equilibrium states and ensembles 136
4.2.3 The elastic continuum limit — the Cauchy-Born rule 138
4.2.4 Non-equilibrium theory 142
4.2.5 Linear response theory and the Green-Kubo formula 144
4.3 Kinetic theory hierarchy 145
4.3.1 The BBGKY hierarchy 145
4.3.2 The Boltzmann equation 147
4.3.3 The equilibrium states 150
4.3.4 Macroscopic conservation laws 153
4.3.5 The hydrodynamic regime 155
4.3.6 Other kinetic models 157
4.4 Electronic structure models 158
4.4.1 The quantum many-body problem 158
4.4.2 Hartree and Hartree-Fock approximation 161
4.4.3 Density functional theory 163
4.4.4 Tight-binding models 168
4.5 Notes 171
Bibliography 172
Chapter 5 Examples of Multi-physics Models 175
5.1 Brownian dynamics models of polymer fluids 176
5.2 Extensions of the Cauchy-Born rule 182
5.2.1 High order, exponential and local Cauchy-Born rules 183
5.2.2 An example of a one-dimensional chain 184
5.2.3 Moving and nanotubes 185
5.3 The shocks contact line problem 188
5.3.1 Classical continuum theory 189
5.3.2 Improved continuum models 190
5.3.3 Measuring the boundary conditions using molecular dynamics 194
5.4 Notes 196
Bibliography 197
Chapter 6 Capturing the Macroscale Behavior 201
6.1 Some classical examples 203
6.1.1 The Car-Parrinello molecular dynamics 203
6.1.2 The quasi-continuum method 206
6.1.3 The kinetic scheme 207
6.1.4 Cloud-resolving convection parametrization 210
6.2 Multigrid and the equation-free approach 210
6.2.1 Extended multigrid method 211
6.2.2 The equation-free approach 212
6.3 The heterogeneous multiscale method 215
6.3.1 The main components of HMM 215
6.3.2 Simulating gas dynamics using molecular dynamics 218
6.3.3 Modifying examples from the HMM viewpoint 220
6.3.4 The classical traditional algorithms to handle multiscale problems 222
6.4 Some general remarks 223
6.4.1 Similarities and differences 223
6.4.2 Difficulties with the three approaches 224
6.5 Seamless coupling 226
6.6 Application to fluids 232
6.7 Stability, accuracy and efficiency 239
6.7.1 The heterogeneous multiscale method 240
6.7.2 The boosting algorithm 243
6.7.3 The equation-free approach 244
6.8 Notes 247
Bibliography 250
Chapter 7 Resolving Local Events or Singularities 255
7.1 Domain decomposition method 256
7.1.1 Energy-based formulation 259
7.1.2 Dynamic atomistic and continuum methods for solids 259
7.1.3 Coupled atomistic and continuum methods for fluids 260
7.2 Adaptive model refinement or model reduction 262
7.2.1 The nonlocal quasicontinuum method 263
7.2.2 Coupled gas dynamic-kinetic models 267
7.3 The heterogeneous multiscale method 270
7.4 Stability issues 271
7.5 Consistency issues illustrated using QC 277
7.5.1 The appearance of the ghost force 277
7.5.2 Removing the ghost force 278
7.5.3 Truncation error analysis 279
7.6 Notes 282
Bibliography 284
Chapter 8 Elliptic Equations with Multiscale Coefficients 288
8.1 Multiscale finite element method 290
8.1.1 The generalized finite element method 290
8.1.2 Residual-free bubbles 292
8.1.3 Multiscale basis functions 293
8.1.4 Multiscale finite volume methods 295
8.1.5 Relations between the various methods 297
8.2 Upscaling via successive elimination of small scale components 298
8.3 Subscaling algorithms 301
8.3.1 Finite element HMM 302
8.3.2 The local microscale problem 304
8.3.3 Error estimates 306
8.3.4 Information about the gradients 307
8.4 Notes 308
Bibliography 313
Chapter 9 Problems with Multiple Time Scales 317
9.1 General setup for time scales 317
9.1.1 The discrete limit theorems 317
9.1.2 Implicit methods 319
9.1.3 Stamatized Runge-Kutta methods 321
9.1.4 HMM 325
9.2 Application of HMM to stochastic simulation algorithms 325
9.3 Coarse-grained molecular dynamics 332
Bibliography 338
Chapter 10 Rare Events 342
10.1 Theoretical background 346
10.1.1 Transition states and reduction to Markov chains 346
10.1.2 Metastable state theory 348
10.1.3 Large deviation theory 350
10.1.4 First exit times 352
10.2 Numerical algorithms 367
10.2.1 Finding transition states 367
10.2.2 Finding the minimal energy path 368
10.2.3 Finding the transition ensemble or the transition tubes 373
10.3 Accelerated dynamics 379
10.3.1 TST-based acceleration techniques 379
10.3.2 Metadynamics 381
10.4 Notes 381
Bibliography 382
Chapter 11 Some Perspectives 385
11.1 Top-down and bottom-up 387
11.2 Problems without scale separation 387
11.2.1 Variational model reduction 388
11.2.2 Modeling memory effects 389
Bibliography 390
Subject Index 390
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