Adnan Khan
Department of Mathematics
Lahore University of Management Sciences

Introduction

Theory of Periodic Homogenization

The Advection Diffusion Equation – Eulerian
and Lagrangian Pictures

Non Standard Homogenization Theory

Summary
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Many physical systems involve more than one
time/space scales
Usually interested in studying the system at
the large scale
Multiscale techniques have been developed for
this purpose
We would like to capture the information at the
fast/small scales in some statistical sense
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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
Heterogeneous Porous Media
 Bhattacharya et.al, Asymptotics of solute dispersion in periodic porous media,
SIAM J. APPL. MATH 49(1):86-98, 1989
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Plasma Physics
 Soward et.al, Large Magnetic Reynold number dynmo action in spatially periodic
flow with mean motion, Proc. Royal Soc. Lond. A 33:649-733

Ocean Atmospheric Science
 Cushman-Roisin et.al, Interactions between mean flow and finite amplitude
mesoscale eddies in a baratropic ocean Geophys. Astrpophys. Fkuid Dynamics
29:333-353, 1984
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Astrophysics
 Knobloch et.al, Enhancement of diffusive transport in Oscillatory Flows, Astroph.
J., 401:196-205, 1992

Fully Developed Turbulence
 Lesieur. M., Turbulence in Fluids, Fluid Mechanics and its Applications 1, Kluwer,
Dordrecht, 1990
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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To smooth out small scale heterogeneities
Assume periodicity at small scales for
mathematical simplification
Capture the behavior of the small scales in
some ‘effective parameter’
Obtain course grained ‘homogenized’ equation
at large scale
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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
As a ‘toy’ problem consider the following
Dirichlet Problem
x


. D(x, )u ( x)   f ( x)



u( x)  g ( x)

x
x  
D is periodic in the second ‘fast’ argument
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Using the ‘ansatz’
1
  x  y

u  u0 ( x, y)  u1 ( x, y)  O( 2 )

Where u i are periodic O(1) functions
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We obtain

2








D




u


f ( x)
y
x
y
x
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Collecting terms with like powers of ε we
obtain the following asymptotic hierarchy
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O(1):  y .D( x, y) yu0   0
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O(ε):  y .D yu1   y D.xu0
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O(ε2):
 y .D yu2   y .(Dxu1 )  x .(D yu1 )  x (Dxu0 )  f ( x)
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Applying periodicity and zero mean conditions
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O(1)  u0 ( x, y)  u0 ( x)
O(ε)  u1 ( x, y)  a( y). xu0  c( x)
where  y .D y ai   i D → The ‘Cell Problem’
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O(ε2)  x .D ( x)xu0 ( x)  f ( x) on  Homogenized
on 
Equation
u0 ( x)  g ( x)
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We have obtained an ‘homogenized’ equation
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The effective diffusivity is given by
x .D ( x)xu0   f ( x)
Dij  D  p ij   D yi a j  p

Where the average over a period is
  p

1
0
  dA
0
a is obtained by solving the ‘cell problem’
 y .D y ai    yi D
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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
Transport is governed by the following
non dimensionalized Advection Diffusion Equation
T 
x t 
1
 V ( x, t )  av( , )   T  aPel T
t 
  

There are different distinguished limits
a  1
a  O(1)
a  1
Weak Mean Flow
Equal Strength Mean Flow
Strong Mean Flow
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We study the simplest case of two scales with
periodic fluctuations and a mean flow
The case of weak and equal strength mean
flows has been well studied
For the strong mean flow case standard
homogenization theory seems to break down
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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For the first two cases we obtain a coarse
grained effective equation
_
_
_
T
*
 V ( x, t )   T    ( K  T ( x, t ))
t


K * is the effective diffusivity given by


Kij  Pe    vi  j 
1
l
ij
is the solution to the ‘cell problem’
The goal is to try an obtain a similar effective
equation for the strong mean flow case
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We study the transport using Monte Carlo
Simulations for tracer trajectories
We compare our MC results to numerics
obtained by extrapolating homogenization
code
We develop a non standard homogenization
theory to explain our results
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We use Monte Carlo Simulations for the
particle paths to study the problem
The equations of motion are given by

x t 
dX1  V1 ( x, t )  av1 ( , ) dt  2aPel1 dW1
  


x t 
dX2  V2 ( x, t )  av2 ( , ) dt  2aPel1 dW2
  


The enhanced diffusivity is given by
K ij 
1
 ( X i (T )  X i (0))( X j (T )  X j (0)) 
2T
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Some MC runs with Constant Mean Flow & CS
fluctuations
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MC and homogenization results agree
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Need a modified Homogenization theory
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We consider one distinguished limit where we
take a  
We develop a Multiple Scales calculation for
the strong mean flow case in this limit
We get a hierarchy of equations (as in standard
Multiple Scales Expansion) of the form
L0 g  f

L0 is the advection operator, f is a smooth function
with mean zero over a cell
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We develop the correct solvability condition for this case

1

g
We want to see if becomes O  large on time scales
 
1
0  t  O 
 

This is equivalent to estimating the following integral
t

 f ( X (s),Y (s))ds
0

The magnitude of this integral will determine the
solvability condition
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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has mean zero over a ‘cell’

f
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Two cases
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
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Low order rational ratio
High Order rational ratio
Magnitude of Integral in
both these cases
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Change to coordinates ‘s’ & ‘t’ along and
perpendicular to the characteristics
Estimate magnitude of the integral in traversing the
cell over the characteristics
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Analysis of the integral gives the following
t


0


C V12  V22
f (s)ds 
 O( )
2 2
 V1 V2
Hence the magnitude of the integral depends on
the ratio V1 of V2 and
For low order rational ratio the integral gets
1
in O( ) time
1
O( )



For higher order
rational ratio the integral stays
1
small over O( ) time

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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We develop the asymptotic expansion in both the
cases
We have the following multiple scales hierarchy
L0T0  0
L0T1  L1T0  0
L0T2  L1T1  L2T0  0
L0T3  L1T2  L2T1  L3T0  0
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We derive the effective equation for the quantity
T  T0  T1   2T2
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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For the low order rational case we get
( R1  R2   2 R3 )T  0
~

1 2
R1   V   x  Pel   v~ ( )  
t
~~
~~
1
1 2
R2  v ( )   x  2 Pel  x    1     2ij  xi  x j  Pel   2     2 v ( )    2 i  2 j  i  j
~
R3  Pe   1  x  Pe  2x  2   v~( )(  x )
1
l

2
x
1
l
2
For the high order rational ratio case we get

1 2 
2
1 2

V



Pe

T


Pe


x
l

l  xT  0
 t

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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Brief exposition of Periodic Homogenization
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Toy Problem to illustrate the process
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Advection Diffusion Equation
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Eulerian Approach – Homogenization
Lagrangian Approach – Monte Carlo Simulation
Non Standard Homogenization Theory
International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010
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