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1/43

Introduction Grad’s Moment System Regularization of Grad’s Moment System Framework of Hyperbolic Moment Method Conclusion

On Moment Model Reduction for KineticEquations

Yuwei Fan

with Zhenning Cai and Ruo Li

School of Mathematical Sciences, Peking University, China

Lunch seminar, SMS-DSEC, Peking UniversityMay 28th, 2015

Yuwei Fan, PKU Moment Model Reduction

2/43


.. Outline

...1 Introduction

...2 Grad’s Moment System

...3 Regularization of Grad’s Moment System

...4 Framework of Hyperbolic Moment Method

...5 Conclusion


3/43

Introduction Grad’s Moment System Regularization of Grad’s Moment System Framework of Hyperbolic Moment Method ConclusionGas Kinetic Theory Model Reduction Hyperbolicity

.. Outline

...1 Introduction




...5 Conclusion


4/43


.. Gas kinetic theory

Kn =Mean free path λ

typical length scale L

0← Kn 10−3 10−2 10−1 1 10

hydrodynamics gas kinetic theory

Euler

equation

N-S equation

no-slip BC.N-S equation slip BC. DSMC?


4/43



0← Kn 10−3 10−2 10−1 1 10


Euler

equation

N-S equation

no-slip BC.N-S equation slip BC. DSMC?

densy gas → rarefied gas → very rarefied gas

↓ ↓ ↓hydrodynamics ? molecular dynamics

↓ ↓ ↓Continuum mechanics → ? ← molecular motion


4/43



densy gas → rarefied gas → very rarefied gas

↓ ↓ ↓hydrodynamics ? molecular dynamics

↓ ↓ ↓Continuum mechanics → ? ← molecular motion

∫Ωx,ξ

f(t,x, ξ) dx dξ = NΩx,ξ.


5/43


.. Boltzmann Equation

Boltzmann equation (Boltzmann 1872) reads:

∂f

∂t+ ξ · ∇xf = Q(f, f),

Q(f, f) is collision term, and (t,x, ξ) ∈ R+ × RD × RD.

Notations:

ρ→ density u→ macroscopic velocity

T → tempurature σij → stress tensor

ρTij = ρT + σij → press tensor qi → heat flux.

Local equilibrium : (Maxwell 1860)

M(t,x, ξ) =ρ(t,x)√

2πT (t,x)D

exp

(−|ξ − u(t,x)|2

2T (t,x)

)


6/43


.. Difficulties in Solving Boltzmann Equation

Boltzmann equation (Boltzmann 1872) reads:

∂f

∂t+ ξ · ∇xf = Q(f, f),

Q(f, f) is collision term, and (t,x, ξ) ∈ R+ × RD × RD.

...1 Collision term Q(f, f) is too complex, e.g. binary collision term:

Q(f, f) =

∫R3

∫S+

(f ′f ′1 − ff1)B(|ξ − ξ1|, σ) dξ1 dn;

...2 High dimension: 1 (t) + 3(x) + 3(ξ) = 7;

...3 ξ ∈ R3.


7/43


.. Model Reduction

0← Kn 10−3 10−2 10−1 1 10


Euler

equation

N-S equation

no-slip BC.N-S equation slip BC. DSMC

Boltzmann

equation

Kn → 0, Boltzmann equation→ Euler equations

Kn ∼ (10−2, 10), Extension of Euler or N-S equations?

Model Reduction: Boltzmann Equation → Macroscopic equations

Grad’s Moment Method

Levermore’s Maximum Entropy Hierarchy

Chapman-Enskog Expansion

etc.


8/43


.. Good Model or Bad Model

Well-posedness of the model:

Hyperbolicity, · · ·Preserving of physics:

Conservation, H-theorem, Galilean invariance, · · ·Approximation efficiency:

# DOF vs Accuracy

Implementation:

BC, Easy to implement,. . .


9/43


.. Hyperbolicity

.Definition (Globally Hyperbolic)..

......

The first-order equations

∂w

∂t+A(w)

∂w

∂x= 0

is globally hyperbolic if the coefficient matrix A(w) is diagonalizablewith real eigenvalues for any admissible w.

.

. What will happen if the system is not hyperbolic?


10/43


.. Hyperbolicity

.Example..

......

The initial value problem

∂

∂t

(uv

)+

(0 a1 0

)∂

∂x

(uv

)= 0,

(u(x, 0)v(x, 0)

)=

(u0(x)v0(x)

).

The characteristic speeds of the system is√a and −

√a, and the system

is hyperbolic if and only if a > 0.This system can be reduced as

utt − auxx = 0,

u(x, 0) = u0(x),

ut(x, 0) = −av0,t(x).

If a is negative, for example a = −1, the system turns to be ellipticequation with two boundary conditions, resulting in the inexistence ofweak solution of the system.


11/43


.. Outline

...1 Introduction




...5 Conclusion


12/43


.. Grad’s Moment Method

Grad’s expansion [Grad CPAM ’1949, Cai, Li SISC ’10]1: expand thedistribution function f around the local equilibriumM:

f(t,x, ξ) =∑

α∈ND

fα(t,x)H[u(t,x),T (t,x)]α (ξ) , (2)

where α = (α1, · · · , αD), H[·]α (·) is defined by

H[u,T ]α (ξ) = (−1)|α| d

α

dξαω[u,T ],

ω[u,T ] =Mρ

=1

√2πT

Dexp

(−|ξ − u|2

2T

),

which are generalized weighted Hermite polynomials.

1We always have

f0 = ρ, fα ≡ 0 if |α| = 1,∑

|α|=1

f2α ≡ 0. (1)


13/43



Grad’s expansion: f =∑

α∈ND

fαH[u,T ]α (ξ)ysubstitute into

Boltzmann Equation:∂f

∂t+ ξ∇f = Q(f, f)ymatching coefficients

Moment Equations:∂w

∂t+

D∑d=1

Ad(w)∂w

∂xd= Qw, w ∈ R∞ytruncation and closure

Grad’s Moment Equations:∂wM

∂t+

D∑d=1

Ad,M (wM )∂wM

∂xd= QwM


14/43


.. Grad’s 13 Moment System

Grad’s 13 moment expansion [Grad 1949]:

f |G13 =M[1 +

Tij − δijT

2T 2

(CiCj − δijC

2)+

2

5

qkρT 2

Ck

(C2

2T− 5

2

)],

where Ci = ξi − ui is the relative velocity.

Substituting the expansion into the Boltzmann equation, and matchingthe coefficients of the polynomials, we can obtain the well-known Grad’s13 moment system


15/43


.. Grad’s 13 Moment System

dρ

dt+ ρ

∂uk

∂xk= 0,

dui

dt+

Tik

ρ

∂ρ

∂xk+

∂Tik

∂xk= 0,

dTij

dt+ 2Tk(i

∂uj)

∂xk+

1

ρ

(4

5

∂q(i

∂xj)+

2

5δij

∂qk∂xk

)= Q(Tij),

dqidt− (TijTjk − 2TTik + T 2δik)

∂ρ

∂xk+

7

5qi∂uk

∂xk+

7

5qk

∂ui

∂xk+

2

5qk

∂uk

∂xi

− ρTik

(∂Tjk

∂xj− 7

6

∂Tjj

∂xk

)+ 2ρT

(∂Tik

∂xk− 1

3

∂Tjj

∂xi

)= Q(qi).


16/43


.. Hyperbolicity of Grad’s 13 Moment System: 1D case

1D case: Grad’s 13 moment system degenerates into

dw

dt+A(w)

∂w

∂x= Q(w) (3)

where w = (ρ, u1, T11, T22, q1)T ,

A(w) =

0 ρ 0 0 0

T11/ρ 0 1 0 00 2T11 0 0 6

5ρ

0 0 0 0 25ρ

−4(T11 − T22)2/9 16q1/5 ρ(11T11 + 16T22)/18 ρ(17T11 − 8T22)/9 0

.

The characteristic polynomial is

det(λI−A) = λ

[λ4 − 2

45(101T11 + 16T22)λ

2

− 96

25

q1ρλ+

1

15(53T 2

11 − 16T11T22 + 8T 222)

],

which only depends on σ11

ρ and q1ρ .

(T11 = T + σ11/ρ, T22 = T − σ11/2ρ)


16/43




dw

dt+A(w)

∂w

∂x= Q(w) (3)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Hyperbolicity region

Non-hyperbolicity region

Maxwellian

Figure 1 : x-axis: q1ρT3/2 , y-axis:

σ11ρT

Hyperbolicity region of (3)


16/43




dw

dt+A(w)

∂w

∂x= Q(w) (3)

Result:(3) is not globally hyperbolic (I. Muller 1998)Maxwellian is an inner point of hyperbolicity region


17/43



3D case: Write Grad’s 13 Moment System in quasi-linear form:

dw

dt+

3∑d=1

Ad∂w

∂xd= Q(w),

where

w = (ρ, u1, u2, u3, T11, T22, T33, T12, T13, T23, q1, q2, q3)T .

It is enough to examine A1 due to the rotational invariance.


18/43



For the Gaussian distribution

f =ρ√

det(2πΘ)exp

(−1

2CTΘ−1C

), Θ =

T T12 0T12 T 00 0 T

,

where |T12| < T , the characteristic polynomial of A1 is

det(λI−A1) =(λ− u1)

3

125[5(λ− u1)

2 − 7T ] · g((λ− u1)

2

T

),

g(x) = 25x4 − 165x3 +

(257 + 48

T 212

T 2

)x2 +

(8T 212

T 2− 105

)x− 28

T 212

T 2.

g(x) has at least one negative zero

⇓A1 has at least two complex eigenvalues

⇓Grad 13 is NOT hyperbolic near Maxwellian


19/43



Hyperbolicity region of Grad 13 moment system on the T12 − q1 plane is:

−1.5 −1 −0.5 0 0.5 1 1.5−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Hyperbolicity region

Non-hyperbolicity region

Maxwellian

Figure 1 : x-axis: q1ρT3/2 , y-axis:

T12T


20/43


.. Hyperbolicity of Grad’s 13 Moment System

1D case:

The system is not globally hyperbolic (I. Muller 1998)

Maxwellian is an inner point of hyperbolicity region

3D case: (Cai, Fan and Li, KRM 2014)

Grad’s 13 moment system is not globally hyperbolic

Maxwellian is NOT an inner point of hyperbolicity region

Question:...1 Why Grad’s moment systems is not globally hyperbolic?...2 Is it possible to regularize Grad’s moment system to be globallyhyperbolic?


21/43


.. Outline

...1 Introduction




...5 Conclusion


22/43



Grad’s expansion: f =∑

α∈ND

fαH[u,T ]α (ξ)ysubstitute into

Boltzmann Equation:∂f

∂t+ ξ∇f = Q(f, f)ymatching coefficients

Moment Equations:∂w

∂t+

D∑d=1

Ad(w)∂w

∂xd= Qw, w ∈ R∞ytruncation and closure

Grad’s Moment Equations:∂wM

∂t+

D∑d=1

Ad,M (wM )∂wM

∂xd= QwM


23/43



1D case (Cai, Fan & Li, CMS ’12):

Moment system:

∂ρ

∂t+ u

∂ρ

∂x+ ρ

∂u

∂x= 0,

ρ∂u

∂t+ T

∂ρ

∂x+ ρu

∂u

∂x+ ρ

∂T

∂x= 0,

1

2ρ∂T

∂t+

1

2ρu

∂T

∂x+ ρu

∂u

∂x+ 3

∂f3∂x

= 0,

∂fα∂t− fα−1

T

ρ

∂ρ

∂x+ (α+ 1)fα

∂u

∂x+

(1

2Tfα−3 +

α− 1

2fα−1

)∂T

∂x

− 3

ρfα−2

∂f3∂x

+ T∂fα−1

∂x+ u

∂fα∂x

+ (α+ 1)∂fα+1

∂x= Sα, α ⩾ 3.

Trucation to give Grad’s (M + 1)-moment system:

discard all the equations containing ∂tfα with α > M ;set fα to be zero for all α > M .


24/43


.. Grad’s Moment Method: Regularization 1D

Grad’s Moment Equations

∂wM

∂t+AM

∂wM

∂x= QMwMy

wM = (ρ, u, T, f3, . . . , fM )T

u ρ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0T/ρ u 1 0 . . . . . . . . . . . . . . . . . 00 2T u 6/ρ 0 . . . . . . . . . . . . . . 00 4f3 ρT/2 u 4 0 . . . . . . . . . 0

−Tf3/ρ 5f4 3f3/2 T u 5 0 . . . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .−TfM−2/ρ MfM−1

12 [(M − 2)fM−2 + TfM−4] −3fM−3/ρ 0 · · · 0 T u M

−TfM−1/ρ (M + 1)fM12 [(M − 1)fM−1 + TfM−3] −3fM−2/ρ 0 . . . . . . 0 T u


24/43




∂wM

∂t+ AM

∂wM

∂x= QMwMy

wM = (ρ, u, T, f3, . . . , fM )T

u ρ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0T/ρ u 1 0 . . . . . . . . . . . . . . . . . 00 2T u 6/ρ 0 . . . . . . . . . . . . . . 00 4f3 ρT/2 u 4 0 . . . . . . . . . 0

−Tf3/ρ 5f4 3f3/2 T u 5 0 . . . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .−TfM−2/ρ MfM−1

12 [(M − 2)fM−2 + TfM−4] −3fM−3/ρ 0 · · · 0 T u M

−TfM−1/ρ 0 − fM−1 +12TfM−3 −3fM−2/ρ 0 . . . . . . 0 T u

The regularization is equivalent to

∂fM+1

∂x= −

(fM

∂u

∂x+

1

2fM−1

∂T

∂x

)


24/43




∂wM

∂t+ AM

∂wM

∂x= QMwM

.Theorem (Cai, Fan & Li, CMS ’12)..

......

The above regularized moment system is globally hyperbolic for anyadmissible wM , and the wave speeds are

u+Cj

√T , (4)

where Cj is the j-th root of Hermite polynomial with order M + 1.


25/43


.. Hyperbolic Regularization: nD

Extension to nD case (Cai, Fan & Li, CPAM ’13):

Grad’s moment system:

∂w

∂t+

D∑d=1

Ad(w)∂w

∂xd= 0.

Regularization:

Ad∂w

∂xd= Ad

∂w

∂xd−

∑|α|=M

RjM (α)IND(α).

⇐⇒ ∂fα+ed

∂xd= −

D∑j=1

(fα+ed−ej

∂uj

∂xd+

1

2fα+ed−2ej

∂T

∂xd

)Regularized moment system:

∂w

∂t+

D∑d=1

Ad(w)∂w

∂xd= 0.


26/43


.. Hyperbolic Regularization: nD

.Theorem (Cai, Fan & Li, CPAM ’13)..

......

The regularized moment system is hyperbolic for any admissible w.Precisely, for a given unit vector n = (n1, · · · , nD), there exists aconstant matrix R partially depending on n that

D∑j=1

njAj(w) = R−1AM (Rw)R,

and this matrix is diagonalizable with eigenvalues as

u · n+Cn,m

√T , 1 ⩽ n ⩽ m ⩽ M + 1,

where Cn,m is the n-th root of the m-th Hermite polynomial.


27/43


.

.What is the essential of the regularization?

.

.Why the regularized moment system is globally hyperbolic?


28/43


.. Outline

...1 Introduction




...5 Conclusion


29/43


.. Observation of Regularization: 1D

Expansion of distribution function:

f =

∞∑α=0

fαH[u,T ]α (ξ) ∈ H[u,T ] := L⟨H[u,T ]

0 ,H[u,T ]1 , · · · ,H[u,T ]

M , · · · ⟩

Grad’s expansion

fGrad =M∑α=0

fαH[u,T ]α (ξ) ∈ H[u,T ]

sub := L⟨H[u,T ]0 ,H[u,T ]

1 , · · · ,H[u,T ]M ⟩

Projection operator

H[u,T ] → H[u,T ]sub

P : f → fGrad


29/43



Projection operator


P : f → fGrad

Substituting the expansion into the Boltzmann equation:

Time derivative:∂Pf∂t

=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

+ fM∂u

∂tH[u,T ]

M+1(ξ) +1

2

∂T

∂t(fM−1H[u,T ]

M+1(ξ) + fMH[u,T ]M+2(ξ))


29/43



Projection operator


P : f → fGrad

Substituting the expansion into the Boltzmann equation:

Time derivative:∂Pf∂t

=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

+XXXXXXXfM∂u

∂tH[u,T ]

M+1(ξ) +1

2

∂T

∂t(XXXXXXXfM−1H[u,T ]

M+1(ξ) +XXXXXXfMH[u,T ]M+2(ξ))

We study the system in the space H[u,T ]sub , so a projection is used

Time derivative:P ∂Pf∂t

=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Corresponding to “Marching coefficients of basis function”


29/43



Projection operator


P : f → fGrad


=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Convection term: ξ∂Pf∂x

=M∑α=0

(⋆Gradα

)H[u,T ]

α (ξ)

+ (⋆1)H[u,T ]M+1(ξ) + (⋆2)H[u,T ]

M+2(ξ) + (⋆3)H[u,T ]M+3(ξ)


29/43



Projection operator


P : f → fGrad


=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Convection term: ξ∂Pf∂x

=M∑α=0

(⋆Gradα

)H[u,T ]

α (ξ)

+XXXXXX(⋆1)H[u,T ]M+1(ξ) +XXXXXX(⋆2)H[u,T ]

M+2(ξ) +XXXXXX(⋆3)H[u,T ]M+3(ξ)

Project it into H[u,T ]sub

Convection term: Pξ ∂Pf∂x

=M∑α=0

(⋆Gradα

)H[u,T ]

α (ξ)


29/43



Projection operator


P : f → fGrad


=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Convection term: Pξ ∂Pf∂x

=M∑α=0

(⋆Gradα

)H[u,T ]

α (ξ)

Marching the coefficient of the basis function H[u,T ]α , α = 0, . . . ,M yields

Grad’s moment system:∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t+ ⋆Grad

α = Q(fα)


30/43



f

Pf

∂Pf

∂t

P∂Pf

∂t

time derivative convection term

pro

jectio

n derivative

pro

jectio

n

f

Pf

∂Pf

∂xξ ∂Pf

∂x

Pξ ∂Pf

∂x

pro

jectio

n

pro

jectio

nderivative

multiply

velocity

Figure 2 : Diagram of the procedure of Grad’s moment system

.

. The procedures of the time derivative and space derivative are different


30/43



f

Pf

∂Pf

∂t

P∂Pf

∂t


pro

jectio

n derivative

pro

jectio

n

f

Pf

∂Pf

∂xξ ∂Pf

∂x

Pξ ∂Pf

∂x

pro

jectio

n

pro

jectio

nderivative

multiply

velocity

Figure 2 : Diagram of the procedure of Grad’s moment systemwwf

Pf

∂Pf

∂t

P∂Pf

∂t


pro

jectio

n derivative

pro

jectio

n

f

Pf

∂Pf

∂x

pro

jectio

n

pro

jectio

nderivative

P∂Pf

∂x

ξP ∂Pf

∂x

PξP ∂Pf

∂x

pro

jectio

n

multiplyvelocity


31/43




=

M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Convection term: ξP ∂Pf∂x

=M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)ξH[u,T ]

α (ξ)

=M∑α=0

(⋆Regularizedα

)H[u,T ]

α (ξ)

+XXXXXX(⋆1)H[u,T ]M+1(ξ) +XXXXXX(⋆2)H[u,T ]

M+2(ξ) +XXXXXX(⋆3)H[u,T ]M+3(ξ)

Project it into the space H[u,T ]sub

PξP ∂Pf∂x

=

M∑α=0

(⋆Regularizedα

)H[u,T ]

α (ξ)


31/43




=M∑α=0

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

Convection term: PξP ∂Pf∂x

=M∑α=0

(⋆Regularizedα

)H[u,T ]

α (ξ)

Marching the coefficient of the basis function H[u,T ]α , α = 0, . . . ,M yields

Regularized moment system:∂fα∂t

+fα−1∂u

∂t+1

2fα−2

∂T

∂t+⋆Regularized

α = Q(fα)


32/43


.

. Why the regularized moment system is globally hyperbolic?


33/43


.. Regularization: 1D

Expansion

f =∞∑

α=0

fαH[u,T ]α (ξ) ≡ ⟨f ,H⟩

Projection

Pf = ⟨P pf ,P bH⟩, P p = P b =(I 0

)∈ R(M+1)×∞

Time derivative

∂f

∂t=

∑α

(∂fα∂t

+ fα−1∂u

∂t+

1

2fα−2

∂T

∂t

)H[u,T ]

α (ξ)

= ⟨D∂w

∂t,H⟩,

w containing all the parameters, such as fα, u, T , and D is a matrix.Convection term

ξ∂f

∂x= ⟨D∂w

∂t, ξH⟩ = ⟨MD

∂w

∂t,H⟩, M is a symmetric matrix


34/43


.. Regularization: 1D

Grad’s moment system:

P pDP Tb

∂P pw

∂t+ P pMDP T

b

∂P pw

∂x= P pQ

Global hyperbolicity requires...1 P pDP T

b is invertible ...2 (P pDP T

b )−1P pMDP T

b is real diagonalizable ×

Regularized moment system:

P pDP Tb

∂P pw

∂t+ P pMP T

b P pDP Tb

∂P pw

∂x= P pQ

Global hyperbolicity requires...1 P pDP T

b is invertible ...2 P pMP T

b is real diagonalizable


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.. Generalized Framework

Kinetic equation∂f

∂t+ v(ξ) · ∇xf = Q(f).

For Boltzmann equation: v(ξ) = ξ

For radiation transport equation: v(ξ) = ξ/|ξ|

...1 Ansatz: Using the general expansion of the distribution function

f(t,x, ξ) =

+∞∑i=0

fi(t,x)φ[η(t,x)]i (ξ).

η = (η1, . . . , ηn), φi is a group orthogonal basis, H[η] ≡ L⟨φ[η]i ⟩.

...2 Projection: Choose a subspace H[η]sub ⊂ H[η], and define a projection

P : H[η] → H[η]sub.

Notations: Denote w by a vector containing all the parametersfα,η.


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∂t+ v(ξ) · ∇xf = Q(f).


f(t,x, ξ) =+∞∑i=0



For Grad’s moment method:

η = (u, T ), φ[η]i (ξ) = H[u,T ]

i (ξ)





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∂t+ v(ξ) · ∇xf = Q(f).


f(t,x, ξ) =+∞∑i=0







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...1 Calculate the time and spatial derivatives of Pf :

∂Pf∂(t, x1, . . . , xD)

...2 Make a projection

P(∂Pf∂s

)= ⟨P pDP T

b

∂P pw

∂s,P bφ

[η]⟩, s = t, x1, . . . , xD.

...3 Calculate convection term and make a projection

P(vk(ξ)P

(∂Pf∂xk

))= ⟨P pMkP

Tb P pDPR

b

∂P pw

∂xk,P bφ

[η]⟩

...4 Match coefficient of basis function

P pDP Tb

∂P pw

∂t+

D∑d=1

P pMdPTb P pDP T

b

∂P pw

∂xd= P pQ(w)


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The above moment system is hyperbolic if...1 P pDP T

b is invertible;

...2 any linear combination of P pMkPTb is real diagonalizable.

.Theorem (Cai, Fan and Li, SIAP 2014, Fan et. al, TR 2014)..

......

Any linear combination of Ak is real diagonalizable if the inner product

of H[η]sub satisfies

⟨vk(ξ)g1, g2⟩[η] = ⟨g1, vk(ξ)g2⟩[η], k = 1, · · · , D, (5)

for any g1, g2 ∈ spanφ[η]i i∈I. Or the projection P is orthogonal

projection(sufficient condition).

In practice, the upper conditions are almost always satisfied.


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.. Properties

The resulting moment system

preserves the conservation of mass, momentum and energy, if

∫RD

1ξ|ξ|2

Pf =

∫RD

1ξ|ξ|2

f ;

Maybe not valid for DVM

is Galilean transformation invariance, iffor any rotation matrix R ∈ RD×D and b ∈ RD,

(Pf)(Rξ + b) = Pf(Rξ + b);

Not valid for DVM

is conservative, if

P ∂Pf∂xd

=∂Pf∂xd

.

Valid for Levermore’s maximum entropy principle


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.. Example 1: anisotropic Hermite moment system

Grad moment method: f ∼M

M(t,x, ξ) =ρ(t,x)√

2πT (t,x)D

exp

(−|ξ − u(t,x)|2

2T (t,x)

)

Anisotropic moment method: f ∼ G

G =ρ

det(2πΘ)exp

(−1

2(ξ − u)TΘ−1(ξ − u)

)Define the basis functions as H[u,Θ]

α (·)

H[u,Θ]α (ξ) = (−1)|α| d

α

dξαω[u,Θ],

ω[u,Θ] = G/ρ.

Ansatz (Fan & Li, SCM 2015):

f =∑α

fαH[u,Θ]α (ξ)


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.. Example 2: Levermore’s Maximum Entropy

Ansatz (Levermore, JSP 1996):

fAnsatz = exp (pn(ξ)) , pn is a polynomial with degree(pn) = n,

such that

∫RD

fAnsatzξα = Fα, |α| ≤ n,

where n is even.


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.. Example 3: Radiative transfer: PN model

Radiative transfer equation is one of the fundamental equations in inertialconfinement fusion (ICF), and it reads

1

c

∂I

∂t+Ω · ∇I = S(ν)− σaI + · · · (6)

where Ω ∈ S+

Equilibrium: Planck function

B(ν, Te) =2hν3

c2

[e

hνkTe − 1

]−1

,

where k is Boltzmann constant.Ansatz:

I(t,x,Ω) =∞∑l=0

l∑m=−l

Iml (t,x)Y ml (Ω).


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.. Outline

...1 Introduction




...5 Conclusion


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.. Conclusion

Grad’s 13 moment system is not hyperbolic even around the localMaxwellian;

It is possible to regularize Grad moment system to be globalhyperbolic;

The essential of the regularization is treating the time and spacederivative in the same way;

A framework for moment model reduction for kinetic equations ispresented;

Thank you for your attention!


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