Weinan E. Principles of Multiscale Modeling

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just chemistry or material science. Indeed, compared with the traditional approach of

focusing on one scale, looking at a problem simultaneously from several diﬀerent scales

and diﬀerent levels of detail is a much more mature way of doing modeling. This does

represent a fundamental change in the way we view modeling.

The multiscale, multi-physics viewpoint opens up unprecedented opportunities for

modeling. It opens up the opportunity to put engineering models on a solid footing.

It allows us to connect engineering applications with basic science. It oﬀers a more

uniﬁed view to mo deling, by focusing more on the diﬀerent levels of physical laws and

the relations b etween them, with the speciﬁc applications as examples. In this way, we

ﬁnd that our approaches to solids and ﬂuids are very much parallel to each other, as we

will explain later in the book.

Despite the exciting opportunities oﬀered by multiscale modeling, one thing we have

learned during the past decade is that we should not expect quick results. Many funda-

mental issues have t o be addressed before its expected impact becomes a reality. These

issues include:

1. A detailed understanding of the relation between the diﬀerent levels of physical

models.

2. Boundary conditions for atomistic models such as molecular dynamics.

3. Systematic and accurate procedures of coarse-graining.

Without properly addressing these and other fundamental issues, we run the risk of

replacing one set of ad hoc models by another, or by ad hoc numerical algorithms. This

is hardly a step forward.

This volume is intended to present a systematic discussion about the basic issues in

multiscale modeling. The emphasis is on the fundamental p rinciples and common issues,

not on speciﬁc applications. Selecting the materials to be covered proves to be a very

diﬃcult task, since the subject is so vast and at the same time, quickly evolving. When

deciding about what should be included in this volume, I have asked the question: What

are the basics that one has to know in order to get a global picture about multiscale

modeling? Since multiscale modeling touches upon almost every aspect of modeling in

almost every scientiﬁc discipline, it is inevitable that some of the discussions are very

terse and some important materials are neglected entirely. Nevertheless, we have tried

to make sure that the basic issues are indeed covered.

x PREFACE

The materials in this volume can be divided into two parts: background materials

and general topics to multiscale modeling. Background materials include:

1. Introduction to the fundamental physics models, ranging from continuum mechan-

ics to quantum mechanics (Chapter 4).

2. Basic analytical techniques for multiscale p roblems, such as averaging methods, ho-

mogenization methods, renormalization group methods, and matched asymptotics

(Chapter 2).

3. Classical numerical techniques that use multiscale ideas (Chapter 3).

For the second part, we have chosen the following topics:

1. Examples of multi-physics models. These are analytical models that use explic-

itly multi-physics coupling (Chapter 5). It is important to realize that multiscale

modeling is not just about developing algorithms, it is also about developing better

physical models.

2. Numerical methods for capturing the macroscopic behavior of complex systems with

the help of microscopic models, in the case when empirical macroscopic models are

inadequate (Chapter 6).

3. Numerical methods for coupling macroscopic and microscopic models lo cally in

order to better resolve localized singularities, defects or other events (Chapter 7).

We also include three more speciﬁc examples: Elliptic partial diﬀerential equations with

multiscale coeﬃcients, problems with slow and fast dynamics and rare events. The ﬁrst

is selected to illustrate the issues with spatial scales. The second and third are selected

to illustrate the issues with time scales.

Some of the background materials have been discussed in numerous textbooks. How-

ever, each one of these textbooks contains materials that are enough for a one-semester

course, and most students and researchers will not b e able to aﬀord the time to go through

them all. Therefore, we have decided to discuss these topics again in a simpliﬁed fashion,

to extract the most essential ideas that are necessary for a basic understanding of the

relevant issues.

The book is intended for scientists and engineers who are interested in modeling and

doing it right. I have tried to strike a balance between precision, which requires some

amount of mathematics and detail, and accessibility, which requires glossing over some of

the details and making compromise on precision. For example, we will discuss asymptotic

analysis quite systematically, but we will not discuss the related theorems. We will always

try to use the simplest examples to illustrate the u nderlying ideas, rather than dwelling

on particular applications.

Many people helped to shape up my view on multiscale modeling. Bj¨orn Engquist

introduced me to the subject while I was a graduate student, at a time when multiscale

modeling was not exactly a fashionable area to work on. I have also beneﬁtted a lot from

the other mentors I have had at diﬀerent stages of my career, they include Alexdandre

Chorin, Bob Kohn, An dy Majda, Stan Osher, George Papanicolaou, and Tom Spencer.

I am very fortunate to have Eric Vanden-Eijnden as a close collaborator. Eric and I

discuss and argue about the issues discussed here so often that I am sure his views

and ideas are reﬂected in many parts of this volume. I would like to thank my other

collaborators: Assyr Abdulle, Carlos Garcia-Cervera, Shanqin Chen, Shiyi Chen, Li-

Tien Cheng, Nick Choly, Tom Hou, Zhongyi Huang, Tiejun Li, Xiantao Li, Chun Liu,

Di Liu, Gang Lu, Jianfeng Lu, Paul Maragakis, Pingbing Ming, Cyrill Muratov, Weiqing

Ren, Mark Robbins, Tim Schulze, Denis Serre, Chi-Wang Shu, Qi Wang, Yang Xiang,

Huanan Yan g, Jerry Yang, Xingye Yue, Pingwen Zhang, and more. I have learned a lot

from them, and it is my good fortune to have the opportun ity to work with them. I

am grateful to the many people that I have consulted with at one point or another on

the issues presented here, including: Achi Brandt, Roberto Car, Emily Carter, Shi Jin,

Tim Kaxiras, Yannis Kervrekidis, Mitch Luskin, Felix Otto, Olof Runborg, Zuowei Shen,

Andrew Stuart, Richard Tsai and Chongyu Wang.

My work has been consistently supported by the Oﬃce of Naval Research. I am

very grateful to Wen Masters and Reza Malek-Madani for their conﬁdence on my work,

particularly at the time when multiscale, multi-physics modeling was not quite as popular

as it is now.

Part of this volume is based on lecture notes taken by my current or former students

Dong Li, Lin Lin, Jianfeng Lu, Hao Shen and Xiang Zhou. Their help is very much

appreciated.

Finally, I would like thank Rebecca Louie and Christina Lipsky for their help in typing

the manuscript.

Weinan E

xii PREFACE

June, 2010

Princeton

Chapter 1

Introduction

1.1 Examples of multiscale problems

Whether we explicitly recognize it or not, multiscale phenomena are part of our daily

lives. We organize our time in days, months and years, as a result of the multiscale

dynamics of the solar system. Our society is organized in a hierarchical structure, from

towns to states, countries and continents. Such a structure h as its historical and political

origin, but it is also a reﬂection of the multiscale geographical structure of the earth.

Moving into the realm of modeling, an important tool for studying functions, signals

or geometrical shapes is to decompose them according to their components at diﬀerent

scales, as is done in Fourier or wavelet expansion. From the viewpoint of physics, all ma-

terials at the microscale are made up of the nuclei and the electrons, whose structure and

dynamics are responsible for the macroscale behavior of the material, such as transport,

wave propagation, deformation and failure.

In fact, it is not an easy task to think of a situation that does not involve any multiscale

characters. Therefore, speaking broadly, it is not incorrect to say that multiscale modeling

encompasses almost every aspect of modeling. However, with such a broad view, it would

be impossible to carry out a serious discussion in any kind of depth . Therefore we will

adopt a narrower viewpoint and focus on a number of issues for which the multiscale

character is the dominating issue and is exploited explicitly in the modeling process.

This includes analytical and numerical techniques that exploit the disparity of scales, as

well as multi-physics problems. Here “multi-physics problems” is perhaps a misnomer,

what we have in mind are problems that involve physical laws at diﬀerent levels of detail,

2 CHAPTER 1. INTRODUCTION

such as quantum mechanics and continuum models. We will start with some simple

examples.

1.1.1 Multiscale data and their representation

A basic multiscale phenomenon comes from the fact that signals (functions, curves,

images) often contain components at disparate scales. One such example is shown in

Figure 1.1 which displays an image that contains large scale edges as well as textures

with small scale features. This observation motivated the decomposition of signals into

diﬀerent components according to their scales. Classical examples of such decomposition

include the Fourier decomposition and wavelet decomposition [21].

Figure 1.1: An image with large scale edges and small scale textures (from TG Planimetric)

1.1.2 Diﬀerential equations with multiscale data

Propagation of wave packets

Consider the wave equation:

∂

u = ∆u (1.1.1)

This is a rather innocent looking diﬀerential equation that describes wave propagation.

Consider now the propagation of a wave packet, which is a solution of (1.1.1) with initial

1.1. EXAMPLES OF MULTISCALE PROBLEMS 3

−4 −3 −2 −1 0 1 2 3 4

−1.5

−1

−0.5

0.5

1.5

Figure 1.2: An example of a wave packet

condition of the form:

u(x, 0) = A(x)e

iS(x)/ε

(1.1.2)

where A, S are smooth functions. As always, we will assume that ε ≪ 1. There are

clearly two scales in this problem: The short wavelength ε and the scale of the envelop

of the wave packet, which is O(1). One can exploit this disparity between the two scales

to ﬁnd simp liﬁed treatment of this problem, as is done in geometric optics. For a review

of the diﬀferent numerical algorithms for treating this kind of problems, we refer to [39].

Mechanics of the composite materials

The mechanical deformation of a material is described by the equations of elasticity

theory:

∇ · τ = 0

τ = λ(∇ · u)I + µ(∇u + (∇u)

) (1.1.3)

where u is the displacement ﬁeld, τ is the stress tensor. The ﬁrst equation describes

force balance. The second equation is the constitutive relation, in which λ and µ are the

Lam´e constants that characterize the material.

To model composite materials, we may simply take

λ(x) = λ

(x), µ(x) = µ

(x) (1.1.4)

where ε is some small parameter that measures the scale of the heterogeneity of the

material. For example, for a two-phase composite (see Figure 1.3), λ and µ take one set

4 CHAPTER 1. INTRODUCTION

of values in one phase and another set of values in the other phase. If the microstructure

happens to be periodic, then we have

(x) = A





, µ

(x) = B





(1.1.5)

where A and B are periodic functions. However, in most cases in practice, the microstruc-

ture tends to be random rather than periodic. A detailed account of such issues can be

found in [44].

Figure 1.3: An example of a two-phase composite material (courtesy of Sal Torquato).

1.1.3 Diﬀerential equations with small parameters

Consider the Navier-Stokes equation for incompressible ﬂows at large Reynolds num-

ber:

(∂

u + (u · ∇)u) + ∇p =

∆u + f, (1.1.6)

∇ · u = 0. (1.1.7)

Here u is the velocity ﬁeld, p is the pressure ﬁeld, ρ

is the (constant) density of the

ﬂuid, and Re ≫ 1 is the Reynolds number. In this example, the highest order derivative

in the diﬀerential equation has a small coeﬃcient. This has some rather important

consequences, for example:

1.2. MULTI-PHYSICS PROBLEMS 5

1. The occurence of boundary layers

2. The occurence of turbulent ﬂows with vortices over a large range of scales.

PDEs of this type, in which the highest order derivatives have small coeﬃcients, are

examples of singular perturbation problems. In most cases, the solutions to such singular

perturbation problems contain features at disparate scales.

1.2 Multi-phy sics problems

1.2.1 Examples of scale-dependent phenomena

We will discuss some well-known examples in which the system response exhibits a

transition as a function of the scales involved.

Black-body radiation

Our ﬁrst example is black-body radiation. Let e

(λ) be the energy density of the

radiation of a black body at temperature T and wavelength λ. Classical statistical

mechanics consideration leads to Rayleigh’s formula

(λ) =

8π

T, (1.2.1)

where k

is the Boltzmann constant. This result ﬁts very well with the experimental

results at long wavelength, but fails drastically at short wavelength. The reason, as was

discovered by Planck, is that at short wavelength, quantum eﬀects become important.

Planck postulated that the energy of a photon has to be an integer multiple of hν where

h is now known as the Planck constant and ν is the frequency of the photon. If quantum

eﬀect is taken into consideration, one arrives at Planck’s formula

(λ) =

8πhc

hc/k

T λ

− 1

(1.2.2)

where c is the speed of light. This result agrees very well with experimental results at

all wavelength [34]. Note that Planck’s formula reduces to Rayleigh’s formula at long

wavelength.

Knudsen number dependence of the heat ﬂux in channel ﬂows

6 CHAPTER 1. INTRODUCTION

Consider gas ﬂow between two parallel plates, one is stationary and the other moves

with a constant speed. The two plates are held at uniform temperature T

and T

respectively. We are interested in the heat ﬂux across the channel as a function of the

Knudsen number, which is the ratio between the mean free path (the average distance

that a gas particle travels before colliding with another gas particle) of the gas particles

and the channel width. What is interesting is that the dependence of the heat ﬂux as

a function of the Knudsen number is non-monotonic: At small Knudsen number, the

heat ﬂux increases as a function of the Knudsen number. At large Knudsen number, it

decreases as a function of the Knudsen number [42]. This phenomenon can be understood

as follows. When the Knudsen number is very small, there is little heat conduction since

the gas is very close t o local equilibrium. In this regime, the dynamics of the gas is

very well described by Euler’s equations of gas dynamics. Heat conduction is a higher

order eﬀect (see Section 4.3). If the Knudsen number is very large, then eﬀectively the

gas particles undergo free streaming. There is little momentum exchange between the

particles since collisions are rare. Hence there is not much heat ﬂux either. In this case,

the dynamics of the gas can be found by solving Boltzmann’s equation with the collision

term neglected (see Section 4.3). We see that the origin for the reduced heat conduction

in the two regimes is very diﬀerent.

The reverse Hall-Petch eﬀect

In the early 1950’s, Hall and Petch independently found that the yield strength of

a material increases as the size of the grains that make up the material decreases [30],

according to the equation:

σ = σ

√

where k is some stren gthening coeﬃcient characteristic to the material, d is the charac-

teristic grain size. Roughly speaking, this is due to the fact that grain boundaries imp ede

dislocation motion.

Recent experiments on many nanocrystalline materials demonstrate that if the grains

reach a critical size which is typically less than 100nm, the yield strength either remains

constant or decreases with decreasing grain size (see Figure 1.4) [20, 41]. This is called the

reverse Hall-Petch eﬀect. The exact mechanism may depend on the speciﬁc material. For

example, in copper, it is suggested that the mechanical response changes from dislocation-

mediated plasticity to grain boundary sliding, as the grain size is decreased to about 10