Computational Transport Through · PDF fileComputational Transport Through Interfaces Shi Jin...

ComputationalTransportThrough
Interfaces
Shi Jin
Introductionto radiativetransferequationwithinterface
Transport-transportcoupling
Transport-diffusioncoupling
conclusions
Computational Transport ThroughInterfaces
Shi JinDepartment of Mathematics,
University of Wisconsin-Madison, USA
joint withXu Yang (Wisconsin>Princeton)
Xiaomei Liao (Harvard)Guangwei Yuan (Beijing)
November 7, 2008

Interfaces
Shi Jin
conclusions
Outline
1 Introduction to radiative transfer equation with interface
2 Transport-transport coupling
3 Transport-diffusion coupling
4 conclusions

Interfaces
Shi Jin
conclusions
Background
radiative transfer equation
Describes the propagation of light and water wavesthrough a turbulent medium.
Semiconductor device modeling and chemotaxismodeling.
Propagation of energy density for linear waves in theheterogeneous media with weak random fluctuation inthe high frequency regime. Spohn, Erds-Yau, Bal-Keller-Papanicolaou-Ryzhik
applications in nano-technology, quantum dots, solarstorage and conversion

Interfaces
Shi Jin
conclusions
The radiative transfer equation
Consider the radiative transfer equation in three-dimension:
t+ v k x = Q(t , x, k),
Q =
(x, k, k)(v |k| v |k|)((t , x, k)(t , x, k)) dk,
x R3, k R3, k = k/|k|, v = 1/.
where the position vector x = (x1, x2, x3) R3, wave vectork = (k1, k2, k3) R3, k is the unit vector in the direction k andv = 1/
is the background sound speed with the density
and the compressibility of the background media. (x, k, k)is the differential scattering cross section and is usuallysymmetric in k and k, and Q is the collision kernel.

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Shi Jin
conclusions
Simplification and rescalingMake simplifications by considering a planar (axial)symmetric medium, then
1v
t+
x=
11
(x , , )[(t , x , )(t , x , )] d,
where = cos , is the angle between the wave movingdirection and x-axis direction; (x , , ) only depends on xand the angle between and .
We introduce a new set of macroscopic variables accordingto t 2t and x x ,
2
v
t+
x=
11
(x , , )[(t , x , )(t , x , )] d,
The hats are dropped for convenience in the following slides.

Interfaces
Shi Jin
conclusions
Simplification and rescalingMake simplifications by considering a planar (axial)symmetric medium, then
1v
t+
x=
11
(x , , )[(t , x , )(t , x , )] d,
where = cos , is the angle between the wave movingdirection and x-axis direction; (x , , ) only depends on xand the angle between and .We introduce a new set of macroscopic variables accordingto t 2t and x x ,
2
v
t+
x=
11
(x , , )[(t , x , )(t , x , )] d,
The hats are dropped for convenience in the following slides.

Interfaces
Shi Jin
conclusions
Asymptotic analysis and diffusion limit
Larsen-Keller, Bensoussan-Lions-Papanicolaou,Bardos-Sentos-Sentis, Golse-J-Levermore
Expanding the solution in powers of ,
(t , x , ) = (0)(t , x , ) + (1)(t , x , ) + 2(2)(t , x , ) +
then using the solvability condition 11
Qd = 0, to the
leading order, (0) is -isotropic and satisfies the diffusionequation.
(0)(t , x) =1
11
(x , , )(0)(t , x , ) d
2v
(0)
t
x(D
(0)
x) = 0

Interfaces
Shi Jin
conclusions
Motivation
We are concerned with the case when the sound speedv contains discontinuities, which generates theinterfaces between different media. transport-transport coupling
Waves crossing this interface will undergo transmissionsand reflections with the energy flux conserved. TheSnells Law holds at the interface.
Fast computation method if the scattering cross section has different scales in two media. transport-diffusion coupling

Interfaces
Shi Jin
conclusions
Motivation
We are concerned with the case when the sound speedv contains discontinuities, which generates theinterfaces between different media. transport-transport couplingWaves crossing this interface will undergo transmissionsand reflections with the energy flux conserved. TheSnells Law holds at the interface.
Fast computation method if the scattering cross section has different scales in two media. transport-diffusion coupling

Interfaces
Shi Jin
conclusions
Geometrical illustration of diffuse transmission

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Shi Jin
conclusions
Interface operators
Transmission operators: R12 and R21.
Reflection operators: R11 and R22.
Response operators: R1 and R2

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Shi Jin
conclusions
Transmission and reflection operators I
For regular reflection and specular transmission:
R11(a)(1) = F11(1)a(1),
R12(a)(1) =1v22v1
F12(2)a(2)
R22(a)(2) = F22(2)a(2),
R21(a)(2) =2v11v2
F21(1)a(1).
The incident ray (1) and the transmission ray (2) satisfiesthe Snells Law of Refraction,
1 21v1
=
1 22v2
.

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Shi Jin
conclusions
Transmission and reflection operators II
For regular reflection and diffuse transmission:
R11(a)(1) = F11(1)a(1),
R12(a)(1) = 2 1
01
1v22v1
F12(2)a(2) d1,
R22(a)(2) = F22(2)a(2),
R21(a)(2) = 2 1
02
2v11v2
F21(1)a(1) d2.

Interfaces
Shi Jin
conclusions
Fresnel reflection-transmission coefficients
Landau-Lifschitz
Assume v1 > v2, then the critical angle is c2 =
1 v22
v21, and
F11(1) =(2v21 1v12)2
(2v21 + 1v12)2,F21(1) =
421v21 21
(1v12 + 2v21)2
F22(2) = F11(1), F12(2) =421v22
22
(1v12 + 2v21)2if2 > c2
F22(2) = 1, F12(2) = 0 if2 c2

Interfaces
Shi Jin

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