Weinan E. Principles of Multiscale Modeling

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BIBLIOGRAPHY 27

[50] C. Zenger, “Sparse grids,” Parallel Algorithms for Partial Diﬀerential Equations,

Kiel, pp. 241–251, Vieweg, Braunschweig, 1991.

[51] R. Zwanzig, “Collision of a gas atom with a cold surface,” J. Chem. Phys., vol. 32,

pp. 1173–1177, 1960.

28 BIBLIOGRAPHY

Chapter 2

Analytical Methods

This chapter is a review of the major analytical techniques that have been developed

for addressing multiscale problems. Our objective is to illustrate the main ideas and to

put things into perspective. It is not our intention to give a thorough treatment of the

topics discussed. Nor is it our aim to provide a rigorous treatment, even though in many

cases rigorous results are indeed available.

It should be remarked that even though most attention on multiscale modeling has

been focused on developing numerical methods, one should not underestimate the use-

fulness of analytical techniques. These analytical techniques oﬀer much needed insight

about the multiscale nature of t he problem. In simple cases, they give rise to simpliﬁed

models. For more complicated problems, they give guidelines for designing and analyzing

numerical algorithms.

The problems that we consider below all have a small parameter that is responsible

for the multiscale behavior. To analyze such problems, our basic strategy is to introduce

additional variables such that in the new variables the scale of interest becomes O(1).

For example, in boundary layer analysis, we introduce an inner variable in which the

width of the boundary layer becomes O(1). We then make an ansatz for the leading

order proﬁle of the multiscale solution, based on our intuition for the problem. If we

have the right intuition, we can use asymptotics to reﬁne our intuition so that better

ansatz can be made. Eventually this process allows us to ﬁnd the leading order behavior

of the multiscale solution. If our intuition is incorrect, the asymptotics will lead us to

nowhere.

30 CHAPTER 2. ANALYTICAL METHODS

2.1 Matched asymptotics

Matched asymptotics is most useful for problems in which the solutions change

abruptly in a small, localized region, usually called the inner region. This small, lo-

calized region can either be boundary layer s or internal layers, or vortices in ﬂuids or

type-II sup erconductors. Matched asymptotics is a systematic procedure for obtaining

approximate solutions for such problems.

The main reason for distinguishing the inner and the outer regions is that the dom-

inant features in the two regions are diﬀerent. To account for the rapid change of the

solution in the inner region, we introduce a stretched variable, also called the inner vari-

able so that in the new variable the scale of the inner region becomes O(1). The dominant

terms in the inner region can be found by transforming the model to the inner variables.

Together with the consideration that the inner solution should b e matched smoothly to

the approximate solution in the outer region often allows us to ﬁnd the leading order

behavior, or the eﬀective models that control the leading order behavior of the solution

in both regions.

2.1.1 A simple advection-diﬀusion equation

Consider the following simple example:

∂

+ ∂

= ε∂

(2.1.1)

for x ∈ [0, 1], with boundary condition:

(0, t) = u

(t), u

(1, t) = u

(t). (2.1.2)

Here as usual, ε is a small parameter. In this problem, the highest order derivative in

the diﬀerential equation has a small coeﬃcient. If we neglect this term, we obtain:

∂

U + ∂

U = 0 (2.1.3)

This is an advection equation. Its solution is called the outer solution. One thing to notice

about the outer solution is that we are only allowed to impose a boun dary condition at

x = 0. In general the boundary condition for u

at x = 1 will not be honored by

solutions of (2.1.3). This suggests that the solutions to (2.1.1) and (2.1.3) m ay behave

2.1. MATCHED ASYMPTOTICS 31

quite diﬀerently near x = 1. Indeed, we will see that for the solution of the original

problem (2.1.1) a boundary layer exists at x = 1 which serves to connect the value of U

near x = 1 to the boundary condition of u

at x = 1.

To see this more clearly, let us change to a stretched variable at x = 1:

y =

x − 1

(2.1.4)

where δ is the unknown boundary layer thickness. In the stretched variable, our original

PDE becomes:

∂

˜u +

∂

˜u =

∂

˜u (2.1.5)

It is easy to see that to be able to obtain some non-trivial balance between the terms in

(2.1.5), we must have δ ∼ ε. Without loss of generality, we will take δ = ε. We then

have, to leading order,

∂

˜u = ∂

˜u (2.1.6)

Integrating once, we obtain

˜u = ∂

˜u + C

(2.1.7)

To be able to match to the outer solution U as we come out of the boundary layer, i.e.

as y → −∞, we must have

lim

y→−∞

∂

˜u = 0 (2.1.8)

Hence, we have

= U(1, t) (2.1.9)

Another integration gives:

˜u(y, t) = U(1, t) + (u

(t) − U(1, t))e

(2.1.10)

or, in terms of the original variable,

˜u

(x, t) = U(1, t) + (u

(t) − U(1, t))e

x−1

(2.1.11)

This gives the leading order behavior inside the boundary layer.

Here we started with the recognition that a boundary layer exists at x = 1. If we had

made the mistake of assuming that the boundary layer was located at x = 0, we would

have obtained the trivial solution ˜u = u

(t) th ere, and we would see that the boundary

condition at x = 1 still presents a problem.

32 CHAPTER 2. ANALYTICAL METHODS

To make this procedure more systematic, we should expand both the inner and outer

solutions ˜u and U, and we should identify the matching region. See [?] for more detailed

discussion of this methodology.

2.1.2 Boundary layers in incompressible ﬂows

Consider two-dimensional incompressible ﬂow in the half plane Ω = {x = (x, y), y ≥

0}. We write the governing equation, the Navier-Stokes equation, as:

∂

+ (u

· ∇)u

+ ∇p

= ε∆u

, (2.1.12)

∇ · u

= 0, (2.1.13)

with no slip boundary condition

(x) = 0, x ∈ ∂Ω (2.1.14)

Here u

= (u

, v

)

is the velocity ﬁeld of the ﬂuid, p

is the pressure ﬁeld. Besides its

practical and theoretical importance, this problem is also historically important since

this was among the earliest major examples treated using boundary layer theory [?].

If we neglect the viscous term at the right hand side, we obtain Euler’s equation

∂

u + (u · ∇)u + ∇p = 0, (2.1.15)

∇ · u = 0. (2.1.16)

In general, solutions to Euler’s equation can only accommodate the boundary condition

that there is no normal ﬂow, i.e. v = 0. Our basic intuition is that when the viscosity ε

is small (ε ≪ 1), the ﬂuid behaves mostly as an inviscid ﬂuid that obeys Euler’s equation

except in a small region next to the boundary in which the viscous eﬀects are important.

Having guessed that a boundary layer exists at y = 0, let us now ﬁnd out the structure

of the solutions inside the boundary layer. As before, we introduce the stretched variable

˜y and perform a change of variable:

˜y = y/δ, and ˜x = x (2.1.17)

The incompressibility condition ∇ · u

= 0 becomes (neglecting the superscript ε)

∂

˜x

u +

∂

˜y

v = 0. (2.1.18)

2.1. MATCHED ASYMPTOTICS 33

To balance the terms, the velocity in the y direction should also be of order δ. Therefore,

we use

˜v = v/δ, ˜u = u, and ˜p = p, (2.1.19)

as our new dependent variables. After this rescaling, (2.1.18) becomes

∂

˜x

u + ∂

˜y

˜v = 0. (2.1.20)

(2.1.12) becomes

∂

˜u + ˜u∂

˜x

˜u + ˜v∂

˜y

˜u + ∂

˜x

˜p = ε∂

˜x

˜u +

∂

˜y

˜u, (2.1.21)

∂

˜v + ˜u∂

˜x

˜v + ˜v∂

˜y

˜v +

∂

˜y

˜p = ε∂

˜x

˜v +

∂

˜y

˜v. (2.1.22)

In the ﬁrst equation, to balance the O(ε/δ

) term with the other terms, we should have

δ ∼ ε

1/2

. This means that the thickness of the boundary layer is O(ε

1/2

). We will take

δ = ε

1/2

. From the second equation, we obtain

∂

˜y

p = 0

This says that to leading order pressure does not change across the boundary layer (i.e.

in the y direction) and is given by the outer solution, the solution to the Euler’s equation.

This is the main simpliﬁcation in the boundary layer.

Retaining only the leading order terms, we obtain Prandtl’s equation governing the

leading order behavior inside the boundary layer (omitting the primes):

∂

u + u∂

u + v∂

u + ∂

P = ∂

u, (2.1.23)

∂

u + ∂

v = 0, (2.1.24)

where the pressure P = P (x, t) is assumed to be given.

Prandtl’s boundary layer theory is derived under the assumption that the ﬂow ﬁeld

can be decomposed into an outer region where viscous eﬀects can be neglected and

an inner region that lies close to the wall. This assumption is valid for laminar ﬂows.

However, it is well-known that at high Reynolds number (in our setting, small ε), ﬂuid

ﬂow generally b ecomes turbulent. In that case, the assumption stated above no longer

holds and Prandtl’s theory ceases to be valid. This does not mean that Prandtl’s theory

is entirely useless. In fact, one most useful application of Prandtl’s equation is in the

34 CHAPTER 2. ANALYTICAL METHODS

analysis of b ound ary layer separation, when the boundary layer detachs from the wall,

convecting the vorticity generated at the boundary to the interior of the ﬂow domain, and

eventually causing the generation of turbulence. This is often attributed to singularity

formation in the solutions of Prandtl’s equation [?]. In addition, an analogous theory

has been developed for turbulent boundary layers [?].

2.1.3 Structure and dynamics of shocks

Viscous shock proﬁle provides one of the simplest examples of an internal layer. For

simplicity, we will use the Burgers equation as a simpliﬁed model of compressible ﬂow

∂

+ ∂



)



= ε∂

. (2.1.25)

As in the last example, the viscosity ε is assu med to be very small. If we set ε = 0, we

obtain the inviscid Burgers equation

∂

u + ∂





= 0. (2.1.26)

As before, this controls the behavior in the outer region.

First let us consider a simple case with initial data

(x, 0) =







1 x < 0,

−1 x > 0,

(2.1.27)

The solution to (2.1.26) with the same initial data is a stationary sh ock

u(x, t) =







1 x < 0,

−1 x > 0.

(2.1.28)

Obviously, this is not a classical solution. The discontinuity at x = 0 is a manifestation

of the fact that we neglected viscous eﬀects. Indeed if we restore the viscous term,

the discontinuity becomes a viscous shock inside which the solution changes rapidly but

remains smooth.

To get an understanding of the detailed structure of the shock, let us introduce the

stretched variable

˜x = x/δ, (2.1.29)

2.1. MATCHED ASYMPTOTICS 35

where δ is the width of the shock. In the new variable ˜x, (2.1.25) becomes (neglecting

the superscript ε)

∂

u +

∂

˜x





∂

˜x

u (2.1.30)

To balance terms, we must have δ ∼ ε. We see that the width of the shock is much smaller

than the width of the boundary layer in incompressible ﬂows (which is O(

√

ε)). This fact

has important implications to the numerical solution of compressible and incompressible

ﬂows at high Reynolds number.

Let δ = ε. The leading order equation now becomes

∂

˜x





= ∂

˜x

u (2.1.31)

with the boundary condition u → ±1 as ˜x → ∓∞. The solution is given by

u = −tan h(˜x/2) = −tanh(x/2ε). (2.1.32)

In the general case of a moving shock, assuming that the shock speed is s, and

introducing

˜x = (x − st)/ε, (2.1.33)

the original PDE (2.1.25) becomes

∂

u −

∂

˜x

u +

∂

˜x





∂

˜x

u. (2.1.34)

The leading order equation is then:

−s∂

˜x

u + ∂

˜x





= ∂

˜x

u. (2.1.35)

Integrating once, we get

−su +

= ∂

˜x

u + C. (2.1.36)

Let u

−

and u

be the values of the inviscid solution at the left and right side of the shock

respectively. In order to match to the inviscid solution, we need:

lim

˜x→±∞

u(˜x, t) = u

(2.1.37)

Therefore, we must have:

−su

−

= −su

= C. (2.1.38)

36 CHAPTER 2. ANALYTICAL METHODS

i.e.

s =

−

+ u

(2.1.39)

This equation determines the speed of the shock. Solving (2.1.36), we obtain th e shock

proﬁle:

u(˜x, t) =

−

˜x

− u

˜x

−

˜x

− e

˜x

−

(2.1.40)

In other words, to leading order, the behavior of u

is given by

˜u

−

x−st

2ε

− u

x−st

2ε

−

x−st

2ε

− e

x−st

2ε

−

(2.1.41)

2.1.4 Transition layers in the Allen-Cahn equation

Next we consider the Allen-Cahn equation, say in two dimension:

∂

= ε∆u

−

′

) (2.1.42)

where V , the free energy, is a double-well potential, with two local minima at u

, u

(see

Figure 2.1.4). This equation describes the dynamics of a scalar order parameter during

ﬁrst order phase transition [?].

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1

Figure 2.1: A double well potential

An inspection of (2.1.42) suggests that the value of u

should be very close to u

and

so that V

′

) ∼ 0, in order for the last term to be balanced by the ﬁrst two terms.