Weinan E. Principles of Multiscale Modeling

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2.4. MULTISCALE EXPANSIONS 57

treated in the begining of this subsection are

= 10

(−x

+ x

), (2.3.89)

= 10

− x

) − x

+ x

, (2.3.90)

= 10

(−x

+ x

) − x

+ x

, (2.3.91)

= 10

− x

). (2.3.92)

In this case, the slow variables are z

= x

+ x

and z

= x

+ x

, and it is easy to see

that the fast process drives the variables toward the ﬁxed point

= x

= z

/2, x

= x

= z

/2, (2.3.93)

meaning that dµ

(x) is atomic:

dµ

(x) = δ(x

− z

/2)δ(x

− z

/2)δ(x

− z

/2)δ(x

− z

/2)dx

. (2.3.94)

Hence from (2.3.87), the limiting dynamics is

= −x

+ x

= (−z

+ z

)/2, (2.3.95)

= −x

+ x

= (z

− z

)/2. (2.3.96)

which from (2.3.86), can also be written in terms of the original variables as

= 0, (2.3.97)

= −x

+ x

, (2.3.98)

= x

− x

, (2.3.99)

= 0. (2.3.100)

A nice modern reference for the averaging methods is [?].

2.4 Multiscale expansions

We now turn into the techniques of multiscale expansion. This is perhaps the most

systematic approach for analyzing problems with disparate scales. We begin with simple

examples of ODEs, and then turn to PDEs and focus on the homogenization technique.

58 CHAPTER 2. ANALYTICAL METHODS

2.4.1 Removing secular terms

Consider

+ y + εy = 0, (2.4.1)

where ε ≪ 1, y(0) = 1 and y

′

(0) = 0. The solution to this problem is given by:

y(t) = cos(

√

1 + εt) = cos



(1 +

+ ···)t



(2.4.2)

If we omit the term εy, the solution would be y(t) = cos t. This is accurate on O(1) time

scale but gives O(1) error on the O(1/ε) time scale.

To analyze the behavior of the solutions over th e O(1/ε) time scale, we use two

time-scale expansion. Let τ = εt be the slow time variable. Assume that

y(t) ∼ y

(t, εt) + εy

(t, εt) + ε

(t, εt). (2.4.3)

Substituting into (2.4.1) and letting τ = εt, we get

∂

∂t

+ y

+ ε



∂

∂t

+ y

+ 2

∂

∂t∂τ

+ y



+ ··· ∼ 0 (2.4.4)

Our objective is to choose functions y

, y

, ··· such that the above equation is valid to

as high an order as possible, when τ = εt. One basic idea of multiscale analysis is to

require that this equation holds for all values of τ, not just when τ = εt. The leading

order equation given by the O(1) terms in (2.4.4) is:

∂

∂t

(t, τ) + y

(t, τ) = 0. (2.4.5)

The solution to this equation can be expressed as y

(t, τ) = C

(τ) cos t + C

(τ) sin t. The

next order equation is

∂

∂t

+ y

= −2

∂

∂t∂τ

− y

. (2.4.6)

Substituting the expression for y

, we get

+ y

= (2C

′

(τ) − C

(τ)) sin t − (2C

′

(τ) + C

(τ)) cos t. (2.4.7)

If the coeﬃcients in front of cos t and sin t do not vanish, then we will have terms that

behave like t cos t in y

, with linear dependence on the fast time t.

2.4. MULTISCALE EXPANSIONS 59

However, when making an ansatz involving two time scales, we always assume that

the lower order terms grow sub linearly as a function of the fast time scale. This is the

minimum condition necessary in order to guarantee that the leading order terms remains

dominant over the time interval of interest. The eﬀective equation for the leading order

terms is often found from this condition. Terms that grow linearly or faster are called

“secular terms”. One example was already shown in the last section when we derived

the averaged equation (2.3.21), as a consequence of the condition that I

(τ, t) grows

sublinearly as a function of τ.

Coming back to (2.4.7), we must require that the coeﬃcients of terms at the right

hand side vanish:

′

(τ) − C

(τ) = 0,

′

(τ) + C

(τ) = 0.

These lead to

(τ) = A

cos(τ/2) + A

sin(τ/2) (2.4.8)

(τ) = −A

sin(τ/2) + A

cos(τ/2) (2.4.9)

Combined with the initial condition, one gets

˜y(t) = cos



εt



cos t − sin



εt



sin t = cos



(1 +



. (2.4.10)

We see that ˜y approximates the original solution accurately over an O(1/ε) time interval.

If we are interested in longer time intervals, we have to extend the asymptotics to higher

order terms. This procedure is called the Poincar´e-Lighthill-Kuo (PLK) metho d [?].

The problem treated here was a simple linear equation whose exact solution can be

written down quite easily. However, it is obvious that the methodology presented is quite

general and can be used for general nonlinear problems.

2.4.2 Homogenization of elliptic equations

Let us consider the problem:

−∇ · (a

(x)∇u

(x)) = f (x), x ∈ Ω (2.4.11)

with the boundary conditon: u

∂Ω

= 0. Here a

(x) = a(x, x/ε), a(x, y) is assumed to be

periodic with respect to y with period Γ = [0, 1]

. We will also assume that a

satisﬁes

60 CHAPTER 2. ANALYTICAL METHODS

the ellipticity condition, uniform in ε, namely that there exists a constant α which is

independent of ε such that

(x) ≥ αI (2.4.12)

where I is the identity tensor.

(2.4.11) is a standard model for many diﬀerent problems in physics and engineering,

including t he mechanical properties of composite materials, heat conduction in hetero-

geneous media, ﬂow in porous media, etc. It has also been used as a standard model

for illustrating some of the most important analytical techniques in multiscale analysis

such as homogenization theory. The standard reference for the material discussed in this

section is [?]. See also [?].

We begin with a formal multiscale analysis with the ansatz:

(x) ∼ u





+ εu





+ ε





+ ··· (2.4.13)

Here the functions u

(x, y), j = 0, 1, 2, ··· are assumed to be periodic with period Γ.

Using

∂

∂x





∼ ∇





∇





+ ··· (2.4.14)

the original PDE (2.4.11) becomes:

∇





∇





(2.4.15)



∇





∇





+ ∇



a∇





+ ∇



a∇





+∇





∇





+ ∇





∇





+∇





∇





+ ∇





∇





+ ··· = −f(x)

To leading order, we have:

−∇





∇





= 0 (2.4.16)

This should hold for y = x/ε. We require that it holds for all values of x and y:

−∇

· (a(x, y)∇

(x, y)) = 0 (2.4.17)

From this, we conclude that

∇

= 0, u

= u

(x) (2.4.18)

2.4. MULTISCALE EXPANSIONS 61

The next order equation is given by:

−∇

· (a(x, y)∇

(x, y)) = ∇

· (a(x, y)∇

(x)) (2.4.19)

This should be viewed as a PDE for u

, the term ∇

enters only as a constant coeﬃcient.

To ﬁnd solutions to this equation, it helps to introduce the cell problem:

−∇

· (a(x, y)∇

) = ∇

· (a(x, y) · e

) (2.4.20)

where e

is the standard j-th basis vector for the Euclidean space R

. (2.4.20) is solved

with the periodic boundary condition. Clearly as the solution to this cell problem, ξ

a function of y. But it also depends on x through the dependence of a on x. We can

then express u

(x, y) = ξ(x, y) · ∇

(x) (2.4.21)

where ξ = (ξ

, . . . , ξ

We now turn to the O(1) equation:

∇

· (a(x, y)∇

(x, y)) + ∇

· (a(x, y)∇

(x, y)) (2.4.22)

+ ∇

· (a(x, y)∇

(x, y)) + ∇

· (a(x, y)∇

(x)) = −f (x)

This is viewed as an equation for u

. To be able to ﬁnd solutions which are periodic in

y, u

and u

must satisfy some solvability condition. To identify this condition, let us

deﬁne the averaging operator (with respect to the fast variable y):

hfi =

Γ=[0,1]

f(y)dy =

f (2.4.23)

Applying this operator to both sides of (2.4.22), we obtain

∇

· (ha(x, y)∇

(x, y)i) + ∇

· (ha(x, y)i∇

(x)) = −f (x) (2.4.24)

Since

a(x, y)∇

(x, y) = a(x, y)∇

(ξ(x, y)∇

(x)), (2.4.25)

we have

∇

· (ha∇

ξ + ai∇

) = −f (2.4.26)

Deﬁne

A(x) = ha(x, y)(I + ∇

ξ(x, y))i (2.4.27)

62 CHAPTER 2. ANALYTICAL METHODS

we arrive at the homogenized equation for u

−∇ · (A(x)∇u

(x)) = f (x) (2.4.28)

Let us look at a simple one-dimensional example. In this case, the cell problem

becomes:

−∂

(a(x, y)∂

ξ) = ∂

a (2.4.29)

Integrating once, we get

a(x, y)(1 + ∂

ξ) = C

(2.4.30)

Dividing by a and applying the averaging operator on both sides, we get

1 =





= C





= C

a(x, y)

dy (2.4.31)

Hence, we obtain

A(x) = C





−1

(2.4.32)

This is the well-known result that the coeﬃcient for the homogenized problem is the

harmonic average rather than the arithmetic average of the original coeﬃcient.

The same result can also be derived by applying multiscale test functions in a weak

formulation for the original PDE. This has the advantage that it can be easily made

rigorous. More importantly, it illustrates how we should think about multiscale problems

through test functions. The basic idea is to use multiscale test functions to probe the

behavior of the multiscale solutions at diﬀerent scales. The basis of this procedure is the

following result of Nguetseng [?]. Early applications of this idea can be found in [?, ?].

Lemma 1. Let {u

} be a sequence of functions, uniformly bounded in L

(Ω), ku

≤ C

for some constant C. Then there exists a subsequence {u

}, and a function u ∈ L

(Ω ×

Γ), Γ = [0, 1]

, such that as ε

→ 0,

(x)ϕ





dx →

u(x, y)ϕ(x, y)dxdy

for any function ϕ ∈ C(Ω × Γ) which is periodic in the second variable with period Γ.

To apply this result, we rewrite the PDE −∇ · (a

(x)∇u

(x)) = f (x) as

= ∇u

, −∇ · (a

(x)w

(x)) = f (x) (2.4.33)

2.4. MULTISCALE EXPANSIONS 63

or in a weak form:

Ω

(x)w

(x)∇ϕ

(x)dx =

Ω

f(x)ϕ

(x)dx (2.4.34)

Applying test function of the form ϕ

(x) = ϕ

(x) + εϕ

(x, x/ε), we obtain





(x)



∇

(x) + ∇





+ ε∇





dx (2.4.35)

f(x)



(x) + εϕ





Using the previous lemma, we conclude that there exists a function w ∈ L

(Ω ×Γ), such

that

a(x, y)w(x, y)(∇

(x) + ∇

(x, y))dxdy =

f(x)ϕ

(x)dx (2.4.36)

Set ϕ

= 0 in this equation, we get

−∇

· hawi = f (2.4.37)

If instead we let ϕ

= 0, we obtain

−∇

· (aw) = 0 (2.4.38)

Since w

= ∇u

, we have

(x)ϕ

(x)dx = −

(x)∇ · ϕ

(x)dx (2.4.39)

Applying the same kind of test function as above, we conclude that there exists a function

u ∈ L

(Ω × Γ) such that

w(x, y)ϕ

(x)dxdy = −

u(x, y)(∇

(x) + ∇

(x, y))dxdy (2.4.40)

Let ϕ

= 0, we get

u(x, y)∇

(x, y)dxdy = 0 (2.4.41)

for any ϕ

. Hence we have

∇

u = 0 (2.4.42)

64 CHAPTER 2. ANALYTICAL METHODS

i.e. u(x, y) = u

(x). Let ϕ

= 0 in (2.4.40), we get

hwi = −∇

hui = −∇

(2.4.43)

Applying the same argument to the equation ∇ × w

= 0, we conclude that there

exists a u

∈ H

(Ω × Γ), such that

w(x, y) = ∇

(x, y) + ∇

(x) (2.4.44)

Substituting into (2.4.37) and (2.4.38), we obtain:







−∇

· (a(∇

+ ∇

)) = 0

−∇

· ha(∇

+ ∇

)i = f

(2.4.45)

It is easy to check that this is equivalent to the homogenized equation obtained earlier.

2.4.3 Homogenization of the Hamilton-Jacobi equations

Consider the following problem

∂

+ H(

, ∇u

) = 0, (2.4.46)

where H(y, p) is assumed to be periodic in y with period Γ = [0, 1]

. Such a problem

arises, for example, as a model for interface propagation in heterogeneous media, where

is the level set function of the interface, H(x/ε, ∇ϕ) = c(x/ε)|∇ϕ| and c(·) is the

given normal velocity.

Homogenization of (2.4.46) was considered in the pioneering, but unpublished paper

by Lions, Papanicolaou and Varadhan [?] (see also [?]). Here we will proceed to get a

quick understanding of the main features of this problem using formal asymptotics.

We begin with the ansatz

(x, t) ∼ u

(x, t) + εu

(x, x/ε, t) + ···

where u

(x, y, t) is periodic in y. The leading order asymptotic equation is given by

∂

+ H(y, ∇

+ ∇

) = 0 (2.4.47)

To understand this problem, we rewrite it as

H(y, ∇

+ ∇

) = −∂

(2.4.48)

2.4. MULTISCALE EXPANSIONS 65

This should be viewed as the cell problem for u

, with p = ∇

as the parameter. Note

that the right-hand-side is indepen dent of y. It should be viewed as a function of the

parameter ∇

. In fact, the following result asserts that it is a well-deﬁned function of

∇

[?]:

Lemma 2. Assume that H(x, p) → ∞ as |p| → ∞ uniformly in x. Then for any

p ∈ R

, there exists a unique constant λ, such that the cell problem

H(y, p + ∇

v) = λ (2.4.49)

has a periodic viscosity solution v.

We refer to [?] for the deﬁnition of viscosity solutions.

This unique value of λ will be denoted as

H(p).

H is the eﬀective Hamiltonian.

Going back to (2.4.48), we obtain the homogenized equation by letting p = ∇

H(∇

) = λ = −∂

(2.4.50)

∂

H(∇

) = 0 (2.4.51)

As the ﬁrst example, let us consider the one-dimensional Hamiltonian H(x, p) =

|p|

/2 −V (x). (In classical mechanics, this would be called the Lagrangian. Here we are

following standard practice in the theory of Hamilton-Jacobi equations and called it the

Hamiltonian [?]). V is assumed to be periodic with period 1 and we normalize V such

that min

V (y) = 0. The eﬀective Hamiltonian

H(p) can be obtained by solving the cell

problem (2.4.49),

|p + ∂

− V (y) = λ (2.4.52)

Obviously, λ ≥ 0. Let us ﬁrst consider the case when (2.4.52) has a smooth solution.

Smooth solutions have to satisfy

p + ∂

v(y) = ±

√

V (y) + λ (2.4.53)

Thus

p = ±

√

V (y) + λdy (2.4.54)

66 CHAPTER 2. ANALYTICAL METHODS

Therefore we can only expect to have smooth solutions when

|p| ≥

√

V (y)dy (2.4.55)

When

|p| ≤

√

V (y)dy (2.4.56)

the solution to the cell problem can not be smooth, i.e. it has to jump between the +

and − branches of (2.4.53). Being a viscosity solution means that u = ∂

v has to satisfy

the entropy condition, which in this case says that the limit of u from the left can not

be smaller than the limit of u from the right:

u(y

+) ≤ u(y

−) (2.4.57)

where y

is any location where a jump occurs. In order to retain periodicity, the two

branches have to intersect. This implies that

λ = 0 (2.4.58)

Indeed, it can be easily veriﬁed that one can always choose appropriate jump locations

to ﬁnd u such that

p + u(y) = ±

√

V (y) (2.4.59)

and the entropy condition is satisﬁed.

In summary, we have

H(p) =







0, |p| ≤

√

V dy,

λ, |p| =

√

V + λ dy, λ ≥ 0.

(2.4.60)

Note that unlike H, the eﬀective Hamiltonian

H(p) is no longer quadratic in p.

As the second example, let us consider the case when H(y, p) = c(y)|p|. The homog-

enized Hamiltonian is then

H(p) = c(y)|∇

v +p|, where v is a periodic viscosity solution

to the cell problem. In th is case, the original Hamiltonian is a homogeneous function of

degree 1 in p. It is easy to see that the homogenized Hamiltonian inherits this property,

i.e.

H(λp) = λ

H(p) for λ > 0. In fact, let v

= λv, then λ

H(p) = c(y)|∇

+ λp| and

is clearly periodic.