Numerical approximation of Knudsen layer for the Euler-Poisson system

E.Canès,N.Crouseilles,H.Guillard,B.Nkonga,andE.Sonnendrüker,Editors

NUMERICAL APPROXIMATION OF KNUDSEN LAYER FOR THE

EULER-POISSON SYSTEM

Fréderique Charles

1

, Niolas Vauhelet

1

, Christophe Besse

2

,

3

, Thierry

Goudon

2

,

3

, Ingrid LaroixViolet

2

,

3

, Jean-Paul Dudon

4

and Laurent

Navoret

5

Abstrat. Inthiswork,weonsidertheomputationoftheboundaryonditionsforthelinearized

EulerPoissonderivedfromtheBGKkinetimodelinthesmallmeanfreepathregime. Boundary

layersaregeneratedfromthefatthattheinomingkinetiuxmightbefarfromthe

thermody-namialequilibrium. In[2℄, theauthors proposeamethodto omputenumeriallythe boundary

onditionsinthehydrodynamilimitrelyingonananalysisoftheboundarylayers. Inthispaper,

wewillextendthesetehniquesintheaseoftheoupledEuler-Poissonsystem.

Résumé. Dansetravail,nousnousintéressonsàl'évaluationnumériquedeonditionsauxlimites

pourlesystèmed'EulerPoissonlinéariséobtenuàpartirdumodèleinétiqueBGKdanslerégime

depetitlibre parours moyen. Des ouhes limites peuvent apparaître en raison du fait que les

uxinétiquesentrantspeuventdiérerdel'équilibrethermodynamique. Laréferene[2℄introduit

uneméthodedealulnumériquedesonditions auxlimitesdansuntel régimehydrodynamique

baséesurl'analysedesouheslimites. Ii,nousétendonsestehniquesauasdusystèmeouplé

d'EulerPoisson.

1. Introdution

Atakinetilevel,astatistialdesriptionofthedynamisofaplasmasubjettoaneletrield

E

(

t, x

)

an be obtained thanks to the Boltzmann equation governing the evolution of the partiles distribution

funtion

F

(

t, x, v

)

. Intheonedimensional framework,thisequationreads:

∂

t

F

+

v∂

x

F

+

qE∂

v

F

=

1

τ

Q

(

F

)

,

(1)

where

q

∈ {−

1

,

1

}

orrespondtothesignofthehargeofpartiles. Herethetimevariable

t

≥

0

,theposition

variable

x

belongs to

(

−

ω, ω

)

⊂

R

and theveloity

v

∈

R

. The parameter

τ

is theKnudsen number;it is

relatedtothemeanfreepathandisassumedtobesmall. Inthiswork,wehoosefortheollisionoperator

Q

theBGKollisionoperator:

Q

(

F

) =

M

(

ρ,u,θ

)

−

F,

wheretheMaxwellian

M

(

ρ,u,θ

)(

v

) =

ρ

√

2

πθ

exp

−

|

v

−

u

|

2

2

θ

,

(2)

1

LaboratoireJaques-LouisLions-UMR7598CNRS&UniversitéPierreetMarieCurie,ParisVI

2

ProjetTeamSIMPAF,INRIALilleNordEurope

3

LaboPaulPainlevéUMR8524,CNRS&USTLille

4

ThalesAleniaSpae,Cannes

5

InstitutMathématiquedeToulouseCNRS&UniversitéPaulSabatier,Toulouse

EDPSienes,SMAI2011

andthemarosopiquantities

(

ρ, u, θ

)

aredened by

density:

ρ

(

t, x

) =

Z

R

F

(

t, x, v

)

dv,

veloity:

ρu

=

Z

R

vF

(

t, x, v

)

dv,

temperature:

ρu

2

₊

_ρθ

₌

Z

R

|

v

|

2

F

(

t, x, v

)

dv.

Inthispaperwerestrittoonsiderasinglespeieofhargedpartiles,thepartilesoftheoppositeharge

beingagivenbakgroundwithaxedandonstantdensity. Theeletrield

E

=

−

∂

x

V

isself-onsistently

denedthroughthePoisson equationsatisedbythepotential

V

:

−

∂

xx

V

=

q

(

ρ

−

1)

.

(3)

Wereallthat

(1

, v,

|

v

|

2

₎

areollisioninvariants:

Z

R





1

v

|

v

|

2





Q

(

F

)

dv

= 0

.

(4)

Itis well-knownthat when

τ

→

0

,thedistribution funtion

F

onvergesto aMaxwellian,whose maro-sopiparameters

ρ

,

u

and

θ

solvestheEulersystem. Thequestionraisedbythisworkisthedetermination

oftheboundaryonditionsonthisEuler systemorrespondingto theonesimposed onthekinetisystem.

Heresystem(1) (3)isompleted withthefollowingboundaryonditions:

γ

inc

F

(

t,

−

ω, v

) =

R

(

γ

out

F

(

t,

−

ω,

·

))(

v

) + Φ

data,L

(

t, v

)

,

for

v >

0

,

(5)

γ

inc

_F

₍

_{t, ω, v}

_{) =}

_R

₍

_γ

out

_F

₍

_{t, ω,}

_·

₎₎₍

_v

_{) + Φ}

data,R

₍

_{t, v}

₎

_,

for

v <

0

,

(6)

where

Φ

data,R

and

Φ

data,L

aregivenfuntions,and

R

isareetionoperator.Forinstaneassuming

R

= 0

and

Φ

data

_{= 0}

orrespondstoafullyabsorbingboundary. Weonsiderheretwotypesofreetionoperator:

thespeularreetionoperatorgivenby

R

(

F

)(

t, x, v

) =

αF

(

t, x,

−

v

)

,

(7)

where the parameter

α

∈

[0

,

1]

represents thefration of reeted partiles, and thereetion operator of

theMaxwelldiuselawgivenby

R

(

F

)(

t,

−

ω, v

) =

α

M

w

(

v

)

Z

w

Z

v

′

<

0

|

v

′

|

F

(

t,

−

ω, v

′

)

dv

′

,

for

v >

0

,

(8)

where

M

w

(

v

) =

ρ

w

√

2

πθ

w

e

−|

v

|

2

/

(2

θ

w

)

,

Z

w

=

Z

v>

0

vM

w

(

v

)

dv,

with

θ

w

the temperature ofthe wall (seee.g. [15℄). Obviously, dealingwith therighthand boundary,

+

ω

replaes

−

ω

andinomingpartileshavenegativeveloities

v <

0

,wehangethesignof

v

′

in(8)aordingly.

Theproblemisalsoompletedwithaninitialondition

F

(0

, x, v

) =

F

init

₍

_{x, v}

₎

_.

ForthePoissonequation, weimposeDirihletboundaryonditions

V

(

t,

−

ω

) =

V

L

,

V

(

t, ω

) =

V

R

.

Thisframeworkmightappearquiterudein omparisontotheomplexboundaryonditionsthatareused

alreadymakeseveraldiultiesappear;weshallgobaktoamoreambitiousmodelelsewhere.

Asthemean freepathgoesto

0

,thekinetimodelanbeapproahedbyasystemof onservationlaws

forwhihthenumberofboundaryonditionsthatshouldbexeddependsonthesolutionitself(seee.g.[9℄).

Infattheboundaryonditionsforthehyperbolisystemdependonthenumberofinomingharateristis

attheboundary. Theaimofthisworkisto presentnumerialmethodstoomputetheboundaryuxesfor

thehydrodynamimodel,andto omparetheresultswithsimulationsofthekinetiequation(1).

Thepaperisorganizedasfollows. InthenextSetion,westudythesimplestasewherewelinearizethe

BoltzmannBGKequationaroundaMaxwelliansteady state. Werstintroduesomenotationsandreall

thetheoretialresultsneeded. Thenwepresentsomenumerialsimulations. InSetion3,weareonerned

with theouplingwith the Poisson equation. Weonsider boththe linearizedproblem and thenon-linear

ase.

2. Analysis for the linearized Euler system

Inthissetion,weareinterestedinthenumerialapproximationoftheboundarylayerforthelinearized

Eulersysteminthease

E

= 0

,i.e. thereisnoouplingwiththePoissonequation.

2.1. Analysis of the boundary layer

Webriey reallhere thetheorialresultsdeveloppedin [9℄ (seealso [13℄). Let us assumethat

ρ

∗

>

0

,

θ

∗

>

0

and

u

∗

∈

R

are given. Weverifyeasily that

M

∗

:=

M

(

ρ

∗

,u

∗

,θ

∗

)

is astationary solutionof (1) with

E

= 0

. Linearizing around the equilibrium

M

∗

in the form

F

=

M

∗

(1 +

f

)

, (1) leads to the linearized

equationfor

f

:

∂

t

f

+

v∂

x

f

=

1

τ

L

∗

(

f

)

,

(9)

where

L

∗

isthelinearizedBGKoperator. Morepreisely,for

f

∈

L

2

₍

_M

∗

dv

)

wehave

L

∗

(

f

) = Π

f

−

f

,with

Π

theorthogonalprojetionof

L

2

₍

_M

∗

dv

)

tothenite dimensional setspanned bytheollisionalinvariant

{

1

, v,

|

v

|

2

_}

(see[2℄):

Π

f

=

ρ

e

ρ

∗

+

v

−

u

∗

θ

∗

e

u

+

|

v

−

u

∗

|

2

θ

∗

−

1

e

θ

2

θ

∗

,

where

(

ρ,

e

u,

e

e

θ

)

aresuh that





e

ρ

e

ρu

∗

+

ρ

∗

u

e

e

ρ

(

u

2

∗

+

θ

∗

) + 2

ρ

∗

u

∗

u

e

+

ρ

∗

θ

e





=

Z

R





1

v

|

v

|

2





f M

∗

dv.

Weanverifythatthefollowingproperties 1

hold:

•

L

∗

isself-adjointfortheinner produtof

L

2

₍

_M

∗

dv

)

,

•

Ker

(

L

∗

) =

Span

{

1

, v,

|

v

|

2

_}

,

•

Ran

(

L

∗

) =

(Ker

(

L

∗

))

⊥

fortheinnerprodutof

L

2

₍

_M

∗

dv

)

,

•

wehavethedissipationproperty

Z

R

L

∗

f f M

∗

dv

≤

0

.

Equation(9)isompletedwiththeboundaryonditionsdeduedfrom(5)(6)

γ

inc

(

M

∗

f

)(

t,

−

ω, v

) =

R

(

γ

out

(

M

∗

f

)(

t,

−

ω,

·

))(

v

) +

φ

data,L

(

t, v

)

,

for

v >

0

,

(10)

γ

inc

(

M

∗

f

)(

t, ω, v

) =

R

(

γ

out

(

M

∗

f

)(

t, ω,

·

))(

v

) +

φ

data,R

(

t, v

)

,

for

v <

0

,

(11)

1

ThesepropertiesobviouslyholdforthelinearizedBGKoperatorbeauseitreduestoamereprojetion;butitisimportant

whereforinstanein thease

v >

0

,wehave

φ

data,L

= Φ

data,L

−

M

∗

+

R

(

M

∗

)

,

andwiththeinitialdata

f

(0

, x, v

) =

f

init

₍

_{x, v}

₎

_.

Formally,when

τ

→

0

thefuntion

f

belongstoKer

(

L

∗

)

andanbethereforedesribedbyaninnitesimal

Maxwellian:

m

_(e

_ρ,

_u,

_e

e

θ

)(

t,x

)

(

v

) =

e

ρ

ρ

∗

+

v

−

u

∗

θ

∗

u

e

+

|

v

−

u

∗

|

2

θ

∗

−

1

e

θ

2

θ

∗

.

(12)

From(9)andthedenition ofKer

(

L

∗

)

wededuethemomentsystem:

∂

t

Z

R





1

v

|

v

|

2





f M

∗

dv

+

∂

x

Z

R

v





1

v

|

v

|

2





f M

∗

dv

= 0

.

Substituting

f

bytheinnitesimalMaxwellian

m

(e

ρ,

u,

e

θ

e

)

leadstothelinearizedEulersystem:

∂

t





e

ρ

e

u

e

θ





+





u

∗

ρ

∗

0

θ

_∗

ρ

∗

u

∗

1

0

2

θ

∗

u

∗





∂

x





e

ρ

e

u

e

θ





= 0

.

(13)

Thesystem(13)writesinmatrixform

∂

t

U

e

+

A∂

x

U

e

= 0

,

(14)

with

A

=





u

∗

ρ

∗

0

θ

∗

ρ

∗

u

∗

1

0

2

θ

∗

u

∗





.

Thissystemisobviouslyhyperboli: thespetrumofthematrix

A

is

{

u

∗

−

√

3

θ

∗

, u

∗

, u

∗

+

√

3

θ

∗

}

. Therefore,

thenumberofboundaryonditionsthat should bexed at theleft(resp. right)boundarydependsonthe

numberofpositive(resp. negative) eigenvalues. Moreover,weandenethefollowingquadratiform

Q

:

U

e

= (

ρ,

_e

u,

_e

θ

e

)

7→

Z

R

v

|

m

_(e

_ρ,

_u,

_e

_θ

_e

₎

|

2

_M

∗

dv.

(15)

Following[2,9℄,weansplitthesetofinnitesimalMaxwelliansaordingtothesignofthequadratiform

Q

:

Ker(

L

∗

) = Λ

+

⊕

Λ

−

⊕

Λ

0

,

where

Λ

±

orresponds to theeigenspaes assoiatedto the positive (resp. negative) eigenvaluesof

A

, and

Λ

0

istheeigenspae assoiatedto 0whenit belongto thespetrumof

A

. Weshalldenote

I

±

_{= dim(Λ}

±

)

thenumberofpositive(resp. negative)eigenvaluesof

A

.

Then, onsidering theleft hand boundary

x

=

−

ω

, itis possibleto split

m

(

ρ,

e

u,

e

e

θ

)

into

m

−

∈

Λ

−

(whih

orresponds to theoutgoingpart) and

m

+

∈

Λ

+

_⊕

_Λ

0

(whih orresponds to theinoming part at the

boundary

x

=

−

ω

):

m

_(e

_ρ,

_u,

_e

θ

e

)

=

m

−

+

m

+

.

(16)

Dealingwith therighthand boundary

x

= +

ω

, therole of

Λ

+

and

Λ

−

is inverted. Ateahboundarythe

outgoingpartisgivenbytheow,whereastheinomingparthastobeimposedasaboundaryonditionto

ompletetheEulersystem.

Theinominguxattheboundariesaredeterminedthankstoaboundarylayeranalysis. Letusonsider

thefollowinghalfspaeproblem

v∂

z

G

=

L

∗

G,

z >

0

, v

∈

R

where

Υ

data

hastobesuitablydened. Wereallthefollowingstatement(see [9℄):

Theorem1. Thereexistsalinearmapping(thatisusuallyalledthe generalizedChandrasekharfuntional)

C

∗

:

L

2

(

R

,

(1 +

|

v

|

)

M

∗

(

v

)

dv

)

→

Λ

+

⊕

Λ

0

Υ

data

_7→

_m

∞

,

where

m

∞

isthelimitas

z

→ ∞

oftheuniquesolution

G

of (17). Moreover,

G

∈

L

∞

₍₀

_,

∞

;

L

2

(

R

, M

∗

(

v

)

dv

))

.

Letusassumethat

f

expandsasfollows

f

=

m

_(e

_ρ,

_u,

_e

e

θ

)

+

G

L

t,

x

+

ω

τ

, v

+

G

R

t,

ω

−

x

τ

, v

+

o(

τ

)

,

where

G

L

and

G

R

standforboundarylayersand aredened asfollows:

•

G

L

₍

_{t, z, v}

_{) =}

_G

₍

_{t, z, v}

₎

with

G

thesolutionof (17)withinomingdata

Υ

data

(

t, v

) =

γ

inc

f

(

t,

−

ω, v

)

−

m

_(e

_ρ,

_u,

_e

e

θ

)

(

t,

−

ω, v

)

,

(18)

andimposing that

lim

z

→

+

∞

G

(

z, v

) = 0

,

(19)

•

G

R

₍

_{t, z, v}

_{) =}

_G

₍

_{t, z,}

₋

_v

₎

with

G

thesolutionof (17)withinomingdata

Υ

data

(

t, v

) =

γ

inc

f

(

t, ω,

−

v

)

−

m

₍

_ρ,

_e

_u,

_e

e

θ

)

(

t, ω,

−

v

)

.

(20)

andondition(19).

Condition (19)expressesthat the boundarylayeris expeted to vanish farfrom theboundary. Using the

deompositionon

Ker(

L

∗

)

in(16),itimpliesthattheunknown

m

+

satises

C

∗

(

γ

inc

f

(

t, ω,

·

)

−

m

−

(

t, ω,

·

)) =

C

∗

(

m

+(

t, ω,

·

))

,

C

∗

(

γ

inc

f

(

t,

−

ω,

·

)

−

m

−

(

t,

−

ω,

·

)) =

C

∗

(

m

+(

t,

−

ω,

·

))

.

(21)

The questionwe addressis onernedwith the numerialapproximationof theoutgoingstate

m

+

, whih

arisesinthedenitionoftheboundaryuxes.

2.2. Numerial resolution of the linearized Euler system

2.2.1. Numerialmethod

Thematrix

A

of thesystem(13)anbediagonalizedunder theform

A

=





u

∗

ρ

∗

0

θ

∗

ρ

∗

u

∗

1

0

2

θ

∗

u

∗





=

P





u

∗

0

0

0

u

∗

+

√

3

θ

∗

0

0

0

u

∗

−

√

3

θ

∗





P

−

1

_,

with

P

=

1

2

√

θ

∗





2

√

θ

∗

ρ

∗

√

3

ρ

∗

−

√

3

ρ

∗

0

3

√

θ

∗

3

√

θ

∗

−

2

√

θ

∗

θ

∗

2

θ

∗

√

3

−

2

θ

∗

√

3





,

P

−

1

₌

1

3

ρ

∗

√

3

θ

∗





2

√

3

θ

∗

0

−

ρ

∗

√

3

/

√

θ

∗

θ

∗

ρ

∗

√

3

θ

∗

ρ

∗

−

θ

∗

ρ

∗

√

3

θ

∗

−

ρ

∗





.

Weset





˜

ρ

diag

˜

u

diag

˜

θ

diag





=

P

−

1





˜

ρ

˜

u

˜

θ

thevariablesinthenewbasisdenedbythetransitionmatrix

P

,andweobtainthreeindependantequations

on

ρ

˜

diag

,

˜

u

diag

,

θ

˜

diag

:



















∂

t

ρ

˜

diag

+

u

∗

∂

x

ρ

˜

diag

= 0

,

(a)

∂

t

u

˜

diag

+ (

u

∗

+

√

3

θ

∗

)

∂

x

u

˜

diag

= 0

,

(b)

∂

t

θ

˜

diag

+ (

u

∗

−

√

3

θ

∗

)

∂

x

θ

˜

diag

= 0

.

()

(22)

Equations(22)should beompleted withboundaryonditions,whihdependonthesignof

u

∗

,

u

∗

+

√

3

θ

∗

and

u

∗

−

√

3

θ

∗

. More preisely, we denote

U

bd,l

= (

ρ

bd,l

, u

bd,l

, θ

bd,l

)

and

U

bd,r

= (

ρ

bd,r

, u

bd,r

, θ

bd,r

)

the

marosopiquantitieswhihhavetobedenedat theboundaries

x

=

−

ω

and

x

=

ω

respetively,andwe set

U

bd,l

=





U

1

bd,l

U

2

bd,l

U

3

bd,l





=

P

−

1

U

bd,l

,

U

bd,r

=





U

1

bd,r

U

2

bd,r

U

3

bd,r





=

P

−

1

U

bd,r

.

Boundaryonditionsofequations(22)are:

˜

ρ

diag

(

−

ω

) =

U

bd,l

1

if

u

∗

>

0

or

ρ

˜

diag

(

ω

) =

U

1

bd,r

if

u

∗

<

0

,

(a)

˜

u

diag

(

−

ω

) =

U

bd,l

2

if

u

∗

+

p

3

θ

∗

>

0

or

u

˜

diag

(

ω

) =

U

2

bd,r

if

u

∗

+

p

3

θ

∗

<

0

,

(b)

˜

θ

diag

(

−

ω

) =

U

bd,l

3

if

u

∗

−

p

3

θ

∗

>

0

or

θ

˜

diag

(

ω

) =

U

3

bd,r

if

u

∗

−

p

3

θ

∗

<

0

.

()

(23)

We then introdue aregular subdivision

{

x

0

, . . . , x

I

+1

}

of the domain

[

−

ω, ω

]

, with

x

i

=

−

ω

+

i

∆

x

and

∆

x

= 2

ω/

(

I

+ 1)

,andatimedisretisation

∆

t

. Equations(22)aresolvednumeriallythanksto anupwind

sheme. Weobtainforinstane,thanksto equation(22)-(a)andboundaryondition(23)-(a),thefollowing

approximations

ρ

˜

n

diag,i

of

ρ

˜

diag

(

n

∆

t, x

i

)

:



























˜

ρ

n

_diag,

+1

₀

=

U

bd,l

1

,n

+1

,

˜

ρ

n

_diag,

+1

₁

= ˜

ρ

n

diag,

1

−

u

∗

∆

t

∆

x

˜

ρ

n

diag,

1

− U

1

,n

bd,l

,

˜

ρ

n

_diag,i

+1

= ˜

ρ

n

diag,i

−

u

∗

∆

t

∆

x

(˜

ρ

n

diag,i

−

ρ

˜

n

diag,i

−

1

)

for

i

∈ {

2

, I

+ 1

}

,

if

u

∗

>

0

,

and



























˜

ρ

n

+1

diag,i

= ˜

ρ

n

diag,i

−

u

∗

∆

t

∆

x

(˜

ρ

n

diag,i

+1

−

ρ

˜

n

diag,i

)

for

i

∈ {

0

, I

−

1

}

,

˜

ρ

n

_diag,I

+1

= ˜

ρ

n

diag,I

−

u

∗

∆

t

∆

x

U

bd,r

1

,n

−

ρ

˜

n

diag,I

,

˜

ρ

n

_diag,I

+1

₊₁

=

U

bd,r

1

,n

+1

,

if

u

∗

<

0

,

where

U

1

,n

bd,l

=

U

bd,l

1

(

n

∆

t

)

and

U

1

,n

bd,r

=

U

bd,r

1

(

n

∆

t

)

. Thesevaluesareomputedbysolving(21). Themethod

tosolvetheseequationsisexplainedinthenextsubsetion. Wedenoteinthesequel

˜

U

i

n

=





˜

ρ

n

i

˜

u

n

i

˜

θ

n

i





=

P





˜

ρ

n

diag,i

˜

u

n

diag,i

˜

θ

n

diag,i





.

(24)

2.2.2. Treatmentofthe boundary onditions

We explain in this setion the method presented in [2℄ to ompute, at eah time step, the

hydrody-namiboundaryonditions

U

n

bd,l

and

U

n

bd,r

fromthe knowledgeof thehydrodynamiquantity

U

˜

n

i

nearthe

boundaries

x

=

±

ω

andtheknowledgeof

φ

data,L

,

φ

data,R

Wefousherethedisussionontheboundary

x

=

−

ω

(thetreatmentoftheboundary

x

=

ω

issimilar).

Weintrodue,at eahtime

t

n

₌

_n

_∆

_t

, theinnitesimalMaxwellian

m

n

bd,l

(

v

) =

ρ

n

bd,l

ρ

∗

+

u

n

bd,l

v

−

u

∗

θ

∗

+

θ

n

bd,l

2

θ

∗

₍

_v

−

u

∗

)

2

θ

∗

−

1

andweassoiateto

m

n

bd,l

itsdeompositionon

Ker

(

L

∗

)

asin(16):

m

n

bd,l

=

m

n

+

+

m

n

−

.

(25)

Theoutgoingpart

m

n

−

∈

Λ

−

isgivenbytheprojetionof

U

˜

n

0

on

Λ

−

:

m

n

−

(

v

) =

X

k

∈

I

−

α

n

k

χ

k

(

v

)

,

α

n

k

=

R

R

v m

U

˜

n

0

χ

k

(

v

)

M

ρ

∗

,u

∗

,θ

∗

(

v

)

dv

R

R

v

|

χ

k

(

v

)

|

2

M

ρ

∗

,u

∗

,θ

∗

(

v

)

dv

,

where

m

˜

U

n

0

isdened in(12). Wedenote

(

χ

0

, χ

1

, χ

2)

anorthogonalbasisof

Ker

(

L

∗

)

whih isheredened

by(see[2℄)

χ

0(

v

) =

√

1

6

|

v

−

u

∗

|

2

θ

∗

−

3

,

χ

1(

v

) =

√

1

6

_√

3

v

√

−

u

∗

θ

∗

+

|

v

−

u

∗

|

2

θ

∗

,

χ

2(

v

) =

√

1

6

_√

3

v

√

−

u

∗

θ

∗

−

|

v

−

u

∗

|

2

θ

∗

,

and

I

+

₌

_k

_{∈ {}

₀

_,

₁

_,

₂

_}

_{, χ}

k

∈

Λ

+

,

I

−

=

k

∈ {

0

,

1

,

2

}

, χ

k

∈

Λ

−

,

I

0

=

k

∈ {

0

,

1

,

2

}

, χ

k

∈

Λ

0

.

Moreover,

Q

(

χ

0) =

ρ

∗

u

∗

,

Q

(

χ

1) =

ρ

∗

(

u

∗

+

p

3

θ

∗

)

,

Q

(

χ

2) =

ρ

∗

(

u

∗

−

p

3

θ

∗

)

,

where

Q

is the quadrati form dened in (15). Consequently, the sign of the eigenvalues

u

∗

−

√

3

θ

∗

,

u

∗

,

u

∗

+

√

3

θ

∗

allowsustoknow

I

−

andthereforetoompute

m

n

−

.

Then we need to ompute

m

n

+

. We rst notie that integrating the half spae problem (17) over the

veloityvariablegives

d

dz

Z

R

v





1

v

|

v

|

2





G

L

(

z, v

)

M

∗

(

v

)

dv

= 0

,

andthen

Z

R

v





1

v

|

v

|

2





G

L

(0

, v

)

M

∗

(

v

)

dv

=

Z

R

v





1

v

|

v

|

2





G

L

(

∞

, v

)

M

∗

(

v

)

dv

= 0

.

(26)

Moreover,we make anapproximationproposed by Maxwell in [19℄ (see also [1,18℄): weassume that the

outgoingdistributionattheboundary

γ

out

_G

L

(0

,

·

)

oinides withthedistribution atinnity,that istosay

γ

out

G

(0

, v

) = lim

z

→

+

∞

G

(

z, v

) = 0

,

v >

0

.

(27)

Then,using(26),(20)and(27),weobtain

Z

v>

0

v





1

v

|

v

|

2





γ

inc

(

f

)

−

m

n

bd,l

Z

v>

0

v





1

v

|

v

|

2





φ

data,L

(

t

n

, v

) +

R

(

M

∗

m

−

n

)(

v

)

−

m

n

−

(

v

)

M

∗

(

v

)

dv

=

Z

v>

0

v





1

v

|

v

|

2





m

n

+

(

v

)

M

∗

(

v

)

− R

(

M

∗

m

n

+

)(

v

)

dv.

(29)

Relation (29) allows us to ompute

m

n

+

. We notie that we havethree equations for

m

n

+

to be satised

whereasdim

(Λ

+

_⊕

_Λ

0

₎

_{∈ {}

₀

_,

₁

_,

₂

_,

₃

_}

. Inthease

u

∗

= 0

i.e. when thesignatureofthequadratiform

Q

is

(1

,

1)

,weusethefatthat wehavetheadditionalonservationlaw(see[2,14℄)

d

dz

Z

R

v

2

χ

0(

v

)

G

(

z, v

)

M

(

ρ

∗

,

0

,θ

∗

)(

v

)

dv

= 0

.

Amethodtosolvethisoverdeterminatedproblemistopikthenumberofequationsneededin(29)(see[14℄).

Here wewill follow insteadthe method in [2℄ where thedetermination of

m

n

+

relies onthe resolution ofa

leastsquaresproblemunderonstraints. Wereferto[2℄forfurther details.

2.3. Numerial omparisons

Theequation (9) is solvedthanksto asplitting sheme : wesolvesuessivelythe freetransport

equa-tion

∂

t

f

+

v∂

x

f

= 0

(with upwind sheme) and the ollision part

∂

t

f

=

1

τ

L

∗

(

f

)

. We notie that this

last equation anbe solvedexatlysine byonservation properties, asolution

f

of thisequation satises

d

dt

R

R

(1

, v, v

2

₎

_f

₍

_v

₎

_dv

_{= 0}

,andtherefore

Π

f

isnotmodiedduringthisstepandanbeonsideredasagiven

soureterm. Wereferto[2℄forfurtherdetails. Inthesimulationpresentedbelow,wetake

∆

v

= 5

·

10

−

4

and

τ

= 10

−

3

,andforbothhydrodynamiandkinetisimulation

ω

= 0

.

5

and

∆

x

= 1

/

500

(inonvenientunits).

Thenaltimeis

t

f

= 0

.

1

andthetimestep

∆

t

isdeterminedbytheCFLondition. Wethenomparethe

numerialsolutionofthelinearized Boltzmannequation(9)withthenumerialsolutionof linearizedEuler

system(14).

OnFigure1,weonsidertheequilibriumstategivenby

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

−

0

.

1

,

1)

,

with speularreetion at boundaries(operator

R

is given by (7)), areetionoeient of

α

= 0

.

1

and

nosoureterm (

φ

data,L

₌

_φ

data,R

_{= 0}

). Inpartiular, thesignature of

Q

is

(1

,

2)

. Inthis ase,resultsare

satisfatory : the density, marosopi veloity and temperature given by theresolution of the linearized

Eulersystemareverylosefrom thosegivenbythelinearizedBoltzmann equation.

Intheseond exampleonFigure2weonsider theequilibrium stategivenby

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

0

,

1)

,

withdiuse reetionatboundaries(operator

R

isgivenby(7)),areetionoeientof

α

= 0

.

8

andno soureterm(

φ

data,L

₌

_φ

data,R

_{= 0}

). Thisis adegenerate ase: the signatureof

Q

ishere

(1

,

1)

. Thus as

notiedin theprevioussetion,wehavefouronservationrelationsandtwoomponentstobendinorder

todene

m

n

+

.

Finallyin thelastexampleongure3wepresenttheaseofnoreetionat boundarieswithnosoure

term(

φ

data,L

₌

_φ

data,R

_{= 0}

). Wehoosethe signature

(3

,

0)

for

Q

. Againin this asetheapproximation

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Density

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Macroscopic velocity

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature

hydrodynamic

kinetic

Figure 1. Comparison betweenthe kinetisimulation and the hydrodynami simulation

at

t

f

= 0

.

1

s,for

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

−

0

.

1

,

1)

,speularreetionatboundariesand

α

= 0

.

1

.

3. Coupling with the Poisson equation

3.1. Analysis of the linearized VlasovPoisson system

Inthis setion, weonsider the linearizationof the oupledBoltzmannPoisson system(1)(3). Let us

assumethat

ρ

∗

>

0

and

θ

∗

>

0

aregiven. Weset

M

∗

(

x, v

) =

ρ

∗

e

−

v

2

2

θ

∗

√

2

πθ

∗

e

−

qV

∗

(

x

)

/θ

∗

R

ω

−

ω

e

−

qV

∗

(

y

)

/θ

∗

dy

=

M

(

ρ

∗

,

0

,θ

∗

)(

v

)

e

−

qV

∗

(

x

)

/θ

∗

R

ω

−

ω

e

−

qV

∗

(

y

)

/θ

∗

dy

.

(30)

Weverifyeasilythat forsuhaMaxwellian,wehave

v∂

x

M

∗

−

q∂

x

V

∗

∂

v

M

∗

= 0

.

(31)

Wehoosefor

V

∗

thesolutionoftheproblem











−

∂

xx

V

∗

=

q

ρ

∗

e

−

qV

_∗

(

x

)

/θ

_∗

R

ω

−

ω

e

−

qV

∗

(

y

)

/θ

∗

dy

−

1

!

,

V

∗

(

−

ω

) =

V

L

,

V

∗

(

ω

) =

V

R

.

(32)

Lemma 1. Assume

V

L

and

V

R

are given in

R

and

ρ

∗

>

0

,

θ

∗

>

0

. Then there existsa unique solution

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Density

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Macroscopic velocity

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature

hydrodynamic

kinetic

Figure 2. Comparison betweenthe kinetisimulation and the hydrodynami simulation

at

t

f

= 0

.

1

s,for

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

0

,

1)

,diusereetionatboundariesand

α

= 0

.

8

.

Proof. Letus dene

V

b

∈

C

2

_([

₋

_{ω, ω}

_])

an extensionoftheboundaryondition:

V

b

(

−

ω

) =

V

L

and

V

b

(

ω

) =

V

R

. This results relies on the remark that a weak solution of (32)is a ritial point in the ane spae

V

b

+

H

0

1

(

−

ω, ω

)

ofthefuntional

J

(

V

) =

1

2

Z

ω

−

ω

|

∂

x

V

|

2

dx

−

ρ

∗

θ

∗

ln

Z

ω

−

ω

e

−

qV /θ

∗

dx

−

Z

ω

−

ω

qV dx.

Therefore,itremainstoprovethatthisfuntionaladmitsauniqueminimizer. Infat,

J

islearlyontinuous andonvexon

H

1

₍

₋

_{ω, ω}

₎

andwehavetheinequality

ln

Z

ω

−

ω

e

−

qV /θ

∗

_dx

≤

k

V

_θ

k

∞

∗

+ ln(2

ω

)

≤

C

k

V

k

H

1

+ ln(2

ω

)

.

Hene,applyingthePoinaréinequality,weget

J

(

V

)

≥ k

V

k

2

H

1

−

C

₀

ρ

_∗

k

V

k

_H

1

−

C

₁

.

Then

J

is oerive and bounded from below on

H

1

₍

₋

_{ω, ω}

₎

. It admits aunique minimizer

V

∗

. Applying

standardtehniques,weanshowthat

V

∗

∈

C

2

₍

₋

_{ω, ω}

₎

.

Thus,thepair

(

M

∗

, V

∗

)

isastationarysolutionoftheBoltzmann-Poissonproblem(1)(3). Welinearize

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Density

hydrodynamic

kinetic

−0.5

0

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Macroscopic velocity

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Temperature

hydrodynamic

kinetic

Figure 3. Comparison betweenthe kinetisimulation and the hydrodynami simulation

at

t

f

= 0

.

1

s,for

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

1

,

0

.

25)

,withnoreetionandnosouretermat

bound-aries.

problem:

∂

t

f

+

v∂

x

f

−

q∂

x

V

∗

∂

v

f

+

q

v

θ

∗

∂

x

V

e

=

1

τ

L

∗

f.

(33)

The linearized ollisionoperator

L

∗

has been dened in the previoussetion. Equation(33)is oupled to

thePoisson equation

−

∂

xx

V

e

=

q

Z

R

f

M

∗

dv,

V

e

(

−

ω

) =

V

e

(

ω

) = 0

.

(34)

Thus, as

τ

goes to

0

thefuntion

f

looks likean innitesimal Maxwellian

m

(e

ρ,

e

u,

e

θ

)

. The properties of

L

∗

desribedaboveyield

Z

R

L

∗

f





1

v

|

v

|

2





M

(

ρ

∗

,

0

,θ

∗

)

dv

= 0

.

Thereforetakingthemomentsof (33),weareledtothefollowingsystem:

∂

t

Z

R





1

v

|

v

|

2





f M

(

ρ

∗

,

0

,θ

∗

)

dv

+

∂

x

Z

R

v





1

v

|

v

|

2





f M

(

ρ

∗

,

0

,θ

∗

)

dv

−

q∂

x

V

∗

Z

R





v

1

|

v

|

2





∂

v

f M

(

ρ

∗

,

0

,θ

∗

)

dv

+

q

∂

x

V

e

θ

∗

Z

R

v





1

v

|

v

|

2



Z

R





1

v

|

v

|

2





∂

v

f M

(

ρ

_∗

,

0

,θ

_∗

)

dv

=

Z

R

v

θ

∗





1

v

|

v

|

2





f M

(

ρ

_∗

,

0

,θ

_∗

)

dv

−

Z

R





0

1

2

v





f M

(

ρ

_∗

,

0

,θ

_∗

)

dv

Substituting

f

by

m

(

ρ,

e

e

u,

θ

e

)

,weusetheidentities

Z

R

m

_(e

_ρ,

_e

_u,

θ

e

)

M

(

ρ

∗

,

0

,θ

∗

)

dv

=

ρ,

e

Z

R

vm

_(e

_ρ,

_u,

_e

θ

e

)

M

(

ρ

∗

,

0

,θ

∗

)

dv

=

ρ

∗

u

e

;

Z

R

v

2

m

_(e

_ρ,

_u,

_e

e

θ

)

M

(

ρ

∗

,

0

,θ

∗

)

dv

=

ρ

∗

θ

e

+

e

ρθ

∗

,

Z

R

v

3

m

_(e

_ρ,

_u,

_e

e

θ

)

M

(

ρ

∗

,

0

,θ

∗

)

dv

= 3

ρ

∗

θ

∗

u.

e

Wededuethat

∂

t





e

ρ

e

u

e

θ





+





0

ρ

∗

0

θ

∗

ρ

∗

0

1

0

2

θ

∗

0





∂

x





e

ρ

e

u

e

θ





=

q∂

x

V

∗







ρ

∗

e

u

θ

∗

e

θ

θ

∗

0







−

q∂

x

V

e





0

1

0





(35)

holds. From(34),thissystemisoupledwiththelinearizedPoissonequation

−

∂

xx

V

e

=

q

ρ

e

e

−

qV

∗

(

x

)

/θ

∗

R

ω

−

ω

e

−

qV

∗

(

y

)

/θ

∗

dy

,

V

e

(

−

ω

) =

V

e

(

ω

) = 0

.

(36)

Likeforthefreeproblem,weassumethat

f

admitsanexpansionoftheform

f

=

m

_(e

_ρ,

_e

_u,

θ

e

)

+

G

L

₍

_t,

x

+

ω

τ

, v

) +

G

R

₍

_t,

ω

−

x

τ

, v

) +

o

(

τ

)

.

Thentheboundarylayers

G

L

and

G

R

satisesthesamehalfspaeproblem(17). Thereforethedetermination

oftheboundaryonditionthatwehavetoimposetosolve(35)followsthetreatmentpresentedintheprevious

setion. Wenotiemoreoverthatweareneessaryinthedegenerateasewherethequadratiformassoiated

tothematrixofthehyperboliequationhasthesignature

(1

,

1)

,i.e.

u

∗

= 0

. Infat,ifwehoosenon-zero

u

∗

,therelation(31)isnotsatisedandtherefore

M

∗

isnotastationarysolutionoftheBoltzmannequation.

WepresentsomeomparisonsbetweenthesimulationofthelinearizedBoltzmann-Poisson equationand

thelinearizedEuler-Poisson system. Inthenumerialproedures weuse,system(32)issolvedthankstoa

xedpointiterativeproedure. Sine

ρ

∗

and

θ

∗

arexed,thepotential

V

∗

is omputedonlyoneforallat

thebeginningofthesimulation. ThelinearizedBoltzmannequation(33)issolvedwiththesameideathan

intheprevioussetion: wesplit bysolvingrstthetransportequationononetimestep

∂

t

f

+

v∂

x

f

−

q∂

x

V

∗

∂

v

f

+

q

v

θ

∗

∂

x

V

e

= 0

,

withanupwindsheme,thenwesolveononetime step

∂

t

f

=

1

τ

L

∗

(

f

)

.

The linearized Euler system(35)is disretized by asplitting method: rstwesolve (35)with zeroat the

righthandsideasdesribedinsubsetion2.2.1and thenwesolveimpliitely

∂

t





e

ρ

e

u

e

θ





=

q∂

x

V

∗







ρ

_∗

u

e

θ

∗

e

θ

θ

∗

0







−

q∂

x

V

e





0

1

0





.

ThePoissonequationisdisretizedthankstoanitedierenemethod. Figures4and5presenttheresults

foralinearizationaroundtheequilibriumstategivenby

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

0

,

0

.

5)

, withasouretermatthe

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.9

1

1.1

1.2

1.3

1.4

1.5

Density

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.05

0

0.05

0.1

0.15

0.2

Macroscopic velocity

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

Temperature

hydrodynamic

kinetic

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Potential

Figure 4. Comparison betweenthe kinetisimulation and the hydrodynami simulation

fortheoupledproblem with

(

ρ

∗

, u

∗

, θ

∗

) = (1

,

0

,

0

.

5)

,noreetionat boundaries(

R

= 0

),

φ

data,L

₌

_m

(2

/

1

.

2

,

0

,

1

.

2

/

2)

,

φ

data,R

₌

_m

(

ρ

∗

,

0

,θ

∗

)

,

q

=

−

1

,

V

L

₌

_V

R

_{= 1}

at

t

f

= 0

.

1

.

3.2. The non-linearEulerPoisson system

Finally,weomebaktothenon-linearproblem(1)(3)presentedin theintrodution. Asthemeanfree

pathgoesto

0

,wegettheEuler-Poissonsystem:



























∂

t

ρ

+

∂

x

(

ρu

) = 0

,

∂

t

(

ρu

) +

∂

x

(

ρu

2

+

ρθ

) =

−

qρ∂

x

V,

∂

t

_ρθ

₊

_ρu

2

2

+

∂

x

₃

2

ρuθ

+

1

2

ρu

3

₌

₋

_qρu∂

x

V,

(37)

oupledto

−

∂

xx

V

=

q

(

ρ

−

1)

.

(38)

Wepresentarstattempttotreatnumeriallytheinitial-boundary-valueproblem,inorporatingthe

ompu-tationofrelevantboundaryonditionsforthehydrodynamis,designedfollowingtheapproahusedforthe

neutralproblemandthelinearized equation. Clearlynonlinearitiesandouplinggiverisetonewdiulties