Asymptotic-Preserving Schemes VS Two-Scale Schemes - Part II

High order TS expansion

0-th order TS-BSL numeri al s heme

Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II

Alexandre MOUTON (CNRS - Lille)

May 25

th

2012 - Porquerolles

Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II



Main ollaborators :

N. Crouseilles (INRIA - Rennes),

E. Frénod (Université de Bretagne Sud - Vannes),

M. Gutni (Université de Strasbourg),

S. Hirstoaga (INRIA - Nan y),

E. Sonnendrü ker (Université de Strasbourg),

...

Main nan ial supports :

INRIA CALVI (CAL ul s ientique et VIsualisation),

INRIA A.E. FUSION,

...




1

High Order Two-S ale expansion of a singularly perturbed onve tion equation

Introdu tion

0-th order Two-S ale expansion

High Order Two-S ale expansion

Con lusions and perspe tives

2

0-th order Two-S ale BSL method for simulating a harged parti le beam

Two-S ale modelization

Semi-Lagrangian method

Numeri al results





Introdu tion




1


Introdu tion




2




Numeri al results





Introdu tion




Motivations

Many physi al phenomena involves several time and spa es s ales.

Magneti Connement Fusion : the presen e of a strong external magneti

eld implies that

The gyro-radius of the parti les around the magneti lines is very small in

front of the rea tor size,

The gyro-period is small in front of the experiment time s ale,

...

Weakly ompressible uids : onsidering a pressure gradient almost null

implies that

The Ma h number is lose to 0,

High speed a ousti waves appear.

Charged parti le beams : the presen e of a highly os illating external

ele tri eld implies

High speed rotations of the beam in the phase spa e,

Small lamentation stru tures in long time experiments,

...




Introdu tion




Motivations : MCF ontext

=⇒ Most of mathemati al models for MCF depend on several physi al

parameters whi h an be very small or very large.

Example (Linear Vlasov equation in the Guiding-Center regime)

∂t

fǫ + v · ∇x

fǫ +

(

E + v ×e||

ǫ

)

· ∇v

fǫ = 0 ,

fǫ(t = 0, x, v) = f

0(x, v) ,

where ǫ is the ratio between the gyro-period of the parti les and the time s ale

of the experiment. In MCF ontext, ǫ ≪ 1.

=⇒ Build a numeri al s heme for approa hing fǫ when ǫ is lose to 0.




Introdu tion




AP S hemes

Main properties :

It is based on a dis retization of the ǫ-dependent problem,

It should apture the limit regime as ǫ → 0,

It has to be stable independently of the amplitude of ǫ.

Many strategies an be used :

Make impli it in time some well- hosen terms,

Re ombination and/or regularization of some equations,

Ma ro-Mi ro de omposition,

...




Introdu tion




Two-S ale Numeri al S hemes

Prin iple :

1 Write an expansion of fǫ of the form

fǫ(t, x, v) ≈N

∑

k = 0

ǫk Fk

(

t,t

ǫ, x, v

)

,

when ǫ is small, and identify a set of equations for the proles

F

k

: [0,T ]× [0, 2π]× R6 → R (N an be equal to 0),

2 Use a numeri al s heme for approa hing F

0

, . . . ,FN

, giving us

F

0,h, . . . ,FN,h on [0,T ]× [0, 2π]× R6

,

We have to solve a set of equations whi h does not depend on ǫ,

3 Approa h fǫ by

fǫ,h(t, x, v) =

N

∑

k = 0

ǫk Fk,h

(

t,t

ǫ, x, v

)

.

Main di ulty : nd some equations for F

0

, . . . ,FN

.




Introdu tion




State-of-the-art

0-th order Two-S ale s hemes :

Plasma physi s : Frénod & Sonnendrü ker (1998, 2000, 2001) ; Frénod &

Mouton (2010) ; Golse & Saint-Raymond (1999, 2003) ; Han-Kwan

(2010,2011,2012) ; ...

Charged parti le beams : Frénod, Raviart & Salvarani (2009) ; Mouton

(2009)

Fluid dynami s : Frénod, Mouton & Sonnendrü ker (2007) ; Ailliot, Frénod

& Montbet (2010)

First order Two-S ale s hemes :

Plasma physi s : Crouseilles, Frénod, Hirstoaga & Mouton (submitted)

Charged parti le beams : Frénod, Gutni & Hirstoaga (submitted)

Fluid dynami s : Ailliot, Frénod & Montbet (2006)

Higher order Two-S ale s hemes :

Plasma physi s : Frénod, Raviart & Sonnendrü ker (2001)

(with numeri al part)




Introdu tion




Main goal

We onsider the following singularly perturbed onve tion equation :

∂t

uǫ(t, x) + Aǫ(t, x) · ∇x

uǫ(t, x) +1

ǫL(t, x) · ∇

x

uǫ(t, x) = 0 ,

uǫ(t = 0, x) = u

0(x) ,

where uǫ = uǫ(t, x) is the solution of the equation, t ∈ [0,T ], x ∈ Rn

(n ≥ 1),

u

0 : Rn → R and Aǫ, L : [0,T ]× Rn → Rn

are given, and where ǫ > 0 is small.

Goal

For N ∈ N given, build a N-th order Two-S ale expansion of uǫ, i.e. approa h

uǫ as follows :

uǫ(t, x) ≈N

∑

k = 0

ǫk Uk

(

t,t

ǫ, x

)

,

and identify a set of equations for U

0

, . . . ,UN

with hypotheses as weak as

possible for L and the sequen e (Aǫ)ǫ> 0

.




Introdu tion




1


Introdu tion




2




Numeri al results





Introdu tion





Prin iple : approa h uǫ(t, x) with

uǫ(t, x) ≈ U

0

(

t,t

ǫ, x

)

,

as ǫ → 0, and identify some equations for U

0

.




Introdu tion




Chara teristi s

We x θ > 0, p ∈ ]1,+∞[, q su h that

1

q

′ +1

q

= 1,

1

p

+ 1

q

′ < 1,

1

q

′ = max( 1q

− 1

n

, 0).

We assume that L ∈ L

∞(

0,T ;(

W

1,∞(Rn))

n

)

,

Given σ ∈ R, x ∈ Rn

, t ∈ [0,T ], we assume that

∂τX(τ ) = L (t,X(τ )) ,X(σ) = x ,

admits a unique θ-periodi solution τ 7−→ X(τ ; x, t;σ).

Additional hypotheses : we assume that L is smooth enough for having

(t, τ, x) 7→ ∂t

X(τ ; x, t; 0) is in(

L

∞(

0,T ; L∞#

(

0, θ;W 1,q(Rn))))

n

,

(t, τ, x) 7→ ∂t

X(τ ; x, t; 0) is in(

L

∞(

0,T ; L∞#

(

0, θ;W 1,∞(Rn))))

n

,

(t, τ, x) 7→ ∇x

X(τ ; x, t; 0) is in(

L

∞(

0,T ; L∞# (0, θ; L∞(Rn))

))

n

2

.




Introdu tion




Chara terization of U

0

Assumptions :

u

0 ∈ L

p(Rn),

(Aǫ)ǫ> 0

is bounded independently of ǫ in(

L

∞(

0,T ;(

W

1,q(Rn))))

n

,

For all t, x, ǫ, ∇x

· L(t, x) = ∇x

· Aǫ(t, x) = 0.

=⇒ Up to a subsequen e, Aǫ two-s ale onverges to A0

= A0

(t, τ, x) in(

L

∞(

0,T ; L∞#

(

0, θ;(

W

1,q(Rn)))))

n

.




Introdu tion





0

Denition

α0

(t, τ, x) = ((∇x

X)(τ ; x, t; 0))−1 (A0

(t, τ,X(τ ; x, t; 0))− (∂t

X)(τ ; x, t; 0)) ,

a

0

(t, x) =1

θ

∫ θ

0

α0

(t, τ, x) dτ .

Theorem

(uǫ)ǫ> 0

two-s ale onverges to U

0

= U

0

(t, τ, x) in L

∞(

0,T ; L∞# (0, θ; Lp(Rn))

)

with

U

0

(t, τ, x) = V

0

(t,X(−τ ; x, t; 0)) , (RB

0

)

where V

0

= V

0

(t, x) ∈ L

∞ (0,T ; Lp(Rn)) is the solution of

∂t

V

0

(t, x) + a

0

(t, x) · ∇x

V

0

(t, x) = 0 ,V

0

(t = 0, x) = u

0(x) .(TS

0

)




Introdu tion




0-th order Two-S ale S heme

1 Apply a lassi al numeri al method for solving (TS

0

) =⇒ nd

V

0,h = V

0,h(t, x),

2 Compute U

0,h(t, τ, x) by dis retizing (RB

0

),

3 Compute the approximation of uǫ given by

uǫ(t, x) ≈ U

0,h

(

t,t

ǫ, x

)

.




Introdu tion




Guiding-Center approximation of the linear Vlasov equation

Linear Vlasov equation :

∂t

fǫ +

(

v

Eǫ + v × Bǫ

)

· ∇(x,v)fǫ +1

ǫ

(

0

v × e||

)

· ∇(x,v)fǫ = 0 ,

fǫ(t = 0, x, v) = f

0(x, v) ,

Eǫ = Eǫ(t, x), Bǫ = Bǫ(t, x), f0 = f

0(x, v) are given, fǫ = fǫ(t, x, v) is theunknown.

Assumptions :

f

0 ∈ L

p(R6),

(Eǫ)ǫ> 0

and (Bǫ)ǫ> 0

are bounded independently of ǫ in(

L

∞(

0,T ;W 1,q(R3)))

3

.

Eǫ → E0

(t, τ, x) ∈(

L

∞(

0,T ; L∞#

(

0, 2π;W 1,q(R3))))

3

two-s ale,

Bǫ → B0

(t, τ, x) ∈(

L

∞(

0,T ; L∞#

(

0, 2π;W 1,q(R3))))

3

two s ale.




Introdu tion




Guiding-Center approximation of the linear Vlasov equation

Theorem

(fǫ)ǫ> 0

two-s ale onverges to F

0

∈ L

∞(

0,T ; L∞#

(

0, 2π; Lp(R6)))

dened by

F

0

(t, x, v) = G

0

(t, x,R(−τ ) v) ,

where R(τ ) =

1 0 0

0 os τ sin τ0 − sin τ os τ

and G

0

= G

0

(t, y, u) in

L

∞(

0,T ; Lp(R6))

solution of

∂t

G

0

(t, y, u) + u|| · ∇y||G

0

(t, y, u)

+ [〈E0

〉 (t, y) + u× 〈B0

〉 (t, y)] · ∇u

G

0

(t, y, u) = 0 ,G

0

(t = 0, y, u) = f

0(y, u) ,

with the notation

〈λ〉 (t, y) =1

2π

∫

2π

0

R(−τ )λ(t, τ, y) dτ .




Introdu tion




Finite Larmor Radius approximation of the linear Vlasov equation

Linear Vlasov equation :

∂t

fǫ +

v||

0

0

Eǫ + v × Bǫ

· ∇(x,v)fǫ +1

ǫ

0

v⊥

v × e||

· ∇(x,v)fǫ = 0 ,

fǫ(t = 0, x, v) = f

0(x, v) ,

Eǫ = Eǫ(t, x), Bǫ = Bǫ(t, x), f0 = f

0(x, v) are given, fǫ = fǫ(t, x, v) is theunknown.

Assumptions :

f

0 ∈ L

p(R6),

(Eǫ)ǫ> 0

and (Bǫ)ǫ> 0

are bounded independently of ǫ in(

L

∞(

0,T ;W 1,q(R3)))

3

.

Eǫ → E0

(t, τ, x) ∈(

L

∞(

0,T ; L∞#

(

0, 2π;W 1,q(R3))))

3

two-s ale,

Bǫ → B0

(t, τ, x) ∈(

L

∞(

0,T ; L∞#

(

0, 2π;W 1,q(R3))))

3

two s ale.




Introdu tion




Finite Larmor Radius approximation of the linear Vlasov equation

Theorem (Case Bǫ = 0)

(fǫ)ǫ> 0

two-s ale onverges to F

0

∈ L

∞(

0,T ; L∞#

(

0, 2π; Lp(R6)))

dened by

F

0

(t, x, v) = G

0

(t, x+ R(−τ ) v,R(−τ ) v) ,

with

R(τ ) =

0 0 0

0 sin τ 1− os τ0 os τ − 1 sin τ

, R(τ ) =

1 0 0

0 os τ sin τ0 − sin τ os τ

,

and G

0

= G

0

(t, y, u) ∈ L

∞(

0,T ; Lp(R6))

is the solution of

∂t

G

0

(t, y, u) + u|| · ∇y||G

0

(t, y, u)

+

[

1

2π

∫

2π

0

(

R(−τ )R(−τ )

)

E0

(t, τ, y + R(τ ) u) dτ

]

· ∇(y,u)G0

(t, y, u) = 0 ,

G

0

(t = 0, y, u) = f

0(y, u) .




Introdu tion




1


Introdu tion




2




Numeri al results





Introdu tion





Goal : build an approximation of uǫ of the form

uǫ(t, x) ≈N

∑

k = 0

ǫk Uk

(

t,t

ǫ, x

)

,

with U

0

, . . . ,UN

satisfying a set of equations independent of ǫ.




Introdu tion




Filtering the 0-th order terms

We x N = 1 =⇒ Establish the 0-th order and rst order two-s ale onvergen e

to some fun tions U

0

and U

1

+ hara terization of these fun tions.

Assumptions :

u

0 ∈ L

p(Rn),

(Aǫ)ǫ> 0


L

∞(

0,T ;(

W

1,q(Rn))))

n

,

For all t, x, ǫ, ∇x

· L(t, x) = ∇x

· Aǫ(t, x) = 0.

Aǫ → A0

= A0

(t, τ, x) in(

L

∞(

0,T ; L∞#

(

0, θ;(

W

1,q(Rn)))))

n

two-s ale.

uǫ → U

0

= U

0

(t, τ, x) in L

∞(

0,T ; L∞# (0, θ; (Lp(Rn)))

)

two-s ale.

What about U

1

?




Introdu tion




Filtering the 0-th order terms

Dene uǫ,1(t, x) =1

ǫ

(

uǫ(t, x)− U

0

(

t,t

ǫ, x

))

.

=⇒

∂t

uǫ,1(t, x) + Aǫ(t, x) · ∇x

uǫ,1(t, x) +1

ǫL(t, x) · ∇

x

uǫ,1(t, x)

=1

ǫ

(

a

0

(

t,t

ǫ, x

)

− Aǫ(t, x)

)

· ∇x

U

0

(

t,t

ǫ, x

)

,

uǫ,1(t = 0, x) = 0 .

Reminder :

a

0

(t, τ, x) = ((∇x

X)(−τ ; x, t; 0))−1 (a0

(t,X(−τ ; x, t; 0))− (∂t

X)(−τ ; x, t; 0)) ,

a

0

(t, x) =1

θ

∫ θ

0

α0

(t, τ, x) dτ ,

α0

(t, τ, x) = ((∇x

X)(τ ; x, t; 0))−1 (A0

(t, τ,X(τ ; x, t; 0))− (∂t

X)(τ ; x, t; 0)) .




Introdu tion





1

Additional hypotheses :

The sequen e (Aǫ,1)ǫ> 0

dened by

Aǫ,1(t, x) =1

ǫ

(

Aǫ(t, x)−A0

(

t,t

ǫ, x

))

,


L

∞(

0,T ;W 1,∞(Rn)))

n

,

L is smooth enough for insuring that (t, τ, x) 7−→ ∂t

X(τ ; x, t; 0) is in(

L

∞(

0,T ; L∞#

(

0, θ;W 1,∞(Rn))))

n

,

The fun tion W

1

dened by

W

1

(t, τ, x) =

∫ τ

0

(a0

(t, x)−α0

(t, σ, x)) · ∇x

V

0

(t, x) dσ ,

is in W

1,∞(

0,T ; L∞#

(

0, θ;W 1,p(Rn)))

.

Aǫ,1 → A1

= A1

(t, τ, x) ∈(

L

∞(

0,T ; L∞#

(

0, θ;W 1,∞(Rn))))

n

two-s ale.




Introdu tion





1

Theorem

(uǫ,1)ǫ> 0

two-s ale onverges to U

1

= U

1

(t, τ, x) inL

∞(

0,T ; L∞# (0, θ; Lp(Rn))

)

dened by

U

1

(t, τ, x) = V

1

(t,X(−τ ; x, t; 0)) +W

1

(t, τ,X(−τ ; x, t; 0)) , (RB

1

)

where V

1

= V

1

(t, x) ∈ L


∂t

V

1

(t, x) + a

0

(t, x) · ∇x

V

1

(t, x) = −1

θ

∫ θ

0

γ1

(t, τ, x) dτ ,

V

1

(t = 0, x) = 0 ,

(TS

1

)

with γ1

dened as

γ1

(t, τ, x) = α1

(t, τ, x) · ∇x

V

0

(t, x)+ ∂t

W

1

(t, τ, x)+α0

(t, τ, x) ·∇x

W

1

(t, τ, x) ,

and α1

dened as

α1

(t, τ, x) = ((∇x

X)(τ ; x, t; 0))−1

A1

(t, τ,X(τ ; x, t; 0)) .




Introdu tion




First order Two-S ale S heme


0

) =⇒ nd

V

0,h = V

0,h(t, x),

2 Compute U


0

),

3 Compute W

1,h from V

0,h,


1

) =⇒ nd

V

1,h = V

1,h(t, x),

5 Compute U


1

),

6 Compute the approximation of uǫ given by

uǫ(t, x) ≈ U

0,h

(

t,t

ǫ, x

)

+ ǫU1,h

(

t,t

ǫ, x

)

.

Remarks :

(TS

0

) and (TS

1

) only dier in the expression of RHS term =⇒ Use same

numeri al method !

Idem for (RB

0

) and (RB

1

).




Introdu tion





We x N ∈ N∗.

Goal : build an approximation of uǫ of the form

uǫ(t, x) ≈N

∑

k = 0

ǫk Uk

(

t,t

ǫ, x

)

,

with U

0

, . . . ,UN

satisfying a set of equations independent of ǫ.




Introdu tion




Hypotheses on (Aǫ)ǫ> 0

Dene re ursively the sequen e (Aǫ,i )ǫ> 0

as

Aǫ,i (t, x) =1

ǫ

(

Aǫ,i−1

(t, x)−Ai−1

(

t,t

ǫ, x

))

, ∀ i = 1, . . . ,N ,

Aǫ,0(t, x) = Aǫ(t, x) ,

Assume that, for all i = 0, . . . ,N,

Aǫ,i → Ai

= Ai

(t, τ, x) ∈(

L

∞ (0,T ; L∞

#

(

0, θ;W 1,∞(Rn))))

n

two-s ale,

Assume that the two-s ale onvergen e of (Aǫ)ǫ> 0

holds in

(

L

∞(

0,T ; L∞#

(

0, θ;W 1,q(Rn))))

n

.

Denition

For all i = 1, . . . ,N,

αi

(t, τ, x) = ((∇x

X)(τ ; x, t; 0))−1

Ai

(t, τ,X(τ ; x, t; 0)) ,

a

i

(t, x) =1

θ

∫ θ

0

αi

(t, τ, x) dτ ,

a

i

(t, τ, x) = ((∇x

X )(−τ ; x, t; 0))−1

a

i

(t,X(−τ ; x, t; 0)) .




Introdu tion




Re urren e hypothesis

Dene re ursively the sequen e (uǫ,i )ǫ> 0

as

uǫ,i (t, x) =1

ǫ

(

uǫ,i−1

(t, x)− U

i−1

(

t,t

ǫ, x

))

, ∀ i = 1, . . . ,N ,

uǫ,0(t, x) = uǫ(t, x) ,

Assume that, for all i = 0, . . . ,N − 1, (uǫ,i )ǫ> 0

two-s ale onverges to

U

i

= U

i

(t, τ, x) in L

∞(

0,T ; L∞# (0, θ; Lp(Rn))

)

.




Introdu tion





0

, . . . ,UN

Theorem (1/2)

We assume that, for any i = 0, . . . ,N − 1, U

i

writes

U

i

(t, τ, x) = V

i

(t,X(−τ ; x, t; 0)) +W

i

(t, τ,X(−τ ; x, t; 0)) . (RB

i

)

with V

0

, . . . ,VN−1

∈ L

∞ (0,T ; Lp(Rn)), and W

0

, . . . ,WN

dened as

W

i

(t, τ, x) =

∫ τ

0

i−1

∑

j = 0

[aj

(t, x)−αj

(t, σ, x)] · ∇x

V

i−1−j

(t, x) dσ

+

∫ τ

0

i−1

∑

j = 0

[aj

(t, x)−αj

(t, σ, x)] · ∇x

W

i−1−j

(t, σ, x) dσ

−∫ τ

0

R

i−1

(t, σ,X(σ; x, t; 0)) dσ ,

R

i

(t, τ, x) = ∂t

U

i

(t, τ, x) +

i

∑

j = 0

a

j

(t, τ, x) · ∇x

U

i−j

(t, τ, x) ,

with the onvention W

0

= R

0

= 0.




Introdu tion





0

, . . . ,UN

Theorem (2/2)

If W

0

, . . . ,WN

∈ W

1,∞(

0,T ; L∞#

(

0, θ;W 1,p(Rn)))

, (uǫ,N)ǫ> 0

two-s ale

onverges to U

N

= U

N

(t, τ, x) ∈ L

∞(

0,T ; L∞# (0,T ; Lp(Rn))

)

with

U

N

(t, τ, x) = V

N

(t,X(−τ ; x, t; 0)) +W

N

(t, τ,X(−τ ; x, t; 0)) . (RB

N

)

For all i = 0, . . . ,N, Vi

= V

i

(t, x) ∈ L


∂t

V

i

(t, x) + a

0

(t, x) · ∇x

V

i

(t, x) = −1

θ

∫ θ

0

γi

(t, τ, x) dτ ,

V

i

(t = 0, x) =

u

0(x) , if i = 0,

0 else.

(TS

i

)

with γi

dened as

γi

(t, τ, x) = ∂t

W

i

(t, τ, x) +α0

(t, τ, x) · ∇x

W

i

(t, τ, x)

+

i

∑

j = 1

αj

(t, τ, x) · [∇x

V

i−j

(t, x) +∇x

W

i−j

(t, τ, x)] .




Introdu tion




N-th order Two-S ale S heme

For all i = 0, . . . ,N, we perform the following omputations :


i

) =⇒ nd

V

i,h = V

i,h(t, x),

2 Compute U

i,h = U

i,h(t, τ, x) by dis retizing (RB

i

) and using W

i,h,

3 Compute R

i,h = R

i,h(t, τ, x) and W

i+1,h = W

i+1,h(t, τ, x),

Having in hands U

0,h, . . . ,UN,h, we ompute the approximation of uǫ given by

uǫ(t, x) ≈ U

0,h

(

t,t

ǫ, x

)

+ ǫU1,h

(

t,t

ǫ, x

)

+ · · ·+ ǫN U

N,h

(

t,t

ǫ, x

)

.

Remarks :

The systems (TS

i

) only dier in the expression of RHS term =⇒ Use

same numeri al method !

Idem for ea h re onstru tion step (RB

i

).




Introdu tion




1


Introdu tion




2




Numeri al results





Introdu tion




Con lusions

A high order Two-S ale S heme has proposed for the singularly perturbed

onve tion equation

∂t

uǫ(t, x) + Aǫ(t, x) · ∇x

uǫ(t, x) +1

ǫL(t, x) · ∇

x

uǫ(t, x) = 0 ,

uǫ(t = 0, x) = u

0(x) ,

This s heme is built re ursively,

The systems (TS

i

) only dier in the expression of the RHS term =⇒ we

an use the same numeri al te hniques for omputing ea h V

i,h ,

These results generalize Frénod, Raviart and Sonnendrü ker's work ( ase

with Aǫ = A).




Introdu tion




Perspe tives

Appli ation to harged parti les beams :

Implement the rst order Two-S ale S heme (see S. Hirstoaga's

presentation),

Compatibility with a oupling with Poisson's equation (see Frénod,

Salvarani & Sonnendrü ker - 2009).

Appli ation to GC and FLR approximations of linear Vlasov equation :

Identify the minimum requirements for Eǫ and Bǫ for building at least a

rst order Two-S ale S heme,

Dis retization of 0-th order and rst order models,

Compatibility with a oupling with Poisson's equation (see Han-Kwan

2010).

More generally :

Repla e L(t, x) by L

(

t,t

ǫ, x

)

in the toy model, with L = L(t, τ, x) being

θ-periodi in τ .






Numeri al results


1


Introdu tion




2




Numeri al results







Numeri al results


Motivations

The nal goal is to develop some AP s hemes for kineti models for

Magneti Connement Fusion,

Multis ale phenomena o ur in this ontext (FLR ee t, ITG instability

et ...) =⇒ Derive some two-s ale limit models,

Need to perform a numeri al validation of these asymptoti models =⇒Two-S ale numeri al methods.

Frénod, M. & Sonnendrü ker - 2007

Frénod, Salvarani & Sonnendrü ker - 2009

=⇒ Develop a Two-S ale numeri al s heme on a Vlasov-type model more

simple than a MCF kineti model for studying the properties of su h s heme.






Numeri al results


Aim

Axisymmetri harged parti le beam kineti model :

∂t

fǫ(t, r , vr ) +v

r

ǫ∂r

fǫ(t, r , vr ) + (Eǫ(t, r) + Ξǫ(t, r)) ∂vr

fǫ(t, r , vr ) = 0 ,

fǫ(t, r , vr ) = f

0(r , vr

) ,1

r

∂r

(r Eǫ(t, r)) =

∫

R

fǫ(t, r , vr ) dvr ,

Ξǫ(t, r) = −1

ǫH

0

r + H

1

(

ω1

t

ǫ

)

r .

The fun tions f

0 : R2 → R, H1

: R → R and the onstant H

0

> 0 are

given,

The unknowns are fǫ = fǫ(t, r , vr ) and Eǫ = Eǫ(t, r).

Goal

Develop a Two-S ale Semi-Lagrangian S heme for approa hing (fǫ,Eǫ) asǫ → 0.






Numeri al results


1


Introdu tion




2




Numeri al results







Numeri al results


Paraxial approximation

Starting point : 3D Vlasov-Maxwell system

∂t

f + v · ∇x

f +q

m

(

E+ v × B

)

· ∇v

f = 0 ,

f (x, v, 0) = f

0(x, v) ,

∇x

· E =ρ

ε0

,

−∇x

× E = ∂t

B ,∇

x

· B = 0 ,

∇x

× B = µ0

J +1

2

∂t

E ,

with

ρ(t, x) = q

∫

R3f (t, x, v) dv , J(t, x) = q

∫

R3v f (t, x, v) dv .






Numeri al results



Assumptions :

Stationary state already rea hed,

The parti le beam is long and thin,

The beam is mono-kineti in z-dire tion,

The beam is axisymmetri ,

Self- onsistent for es are negle ted in longitudinal dire tion z ,

The external ele tri eld is l -periodi in z-dire tion and does not depend

on t,

The angular momentum of the beam is null at the sour e of the beam.






Numeri al results



3D Vlasov-Maxwell model is redu ed to its paraxial approximation :

∂z

f +v

v

z

· ∇x

f +q

m v

z

(

Ξ+ E

)

· ∇v

f = 0 ,

f (x, z = 0, v) = f

0(x, v) ,

−∇x

φ = E , −∆x

φ =q

ε0

∫

R2

f dv ,

where Ξ is the external ele tri eld of the form

Ξ(x, z) = −H

0

x+ H

1

(

ω1

z

l

)

x ,

with H

0

> 0 is a positive onstant tension, H

1

is a l -periodi tension, and

where ω1

is a xed dimensionless onstant.

Degond & Raviart - 1993

Filbet & Sonnendrü ker - 2006






Numeri al results


Res aling

The parti le beam is long and thin

=⇒Char. length in x-dire tion

Char. length in z-dire tion

= ǫ ,

Assume that l is small in front of the length of the parti le a elerator

=⇒l

Char. length in z-dire tion

= ǫ .

Res aled model

∂t

fǫ +1

ǫv · ∇

x

fǫ + (Ξǫ + Eǫ) · ∇v

fǫ = 0 ,

fǫ(t = 0, x, v) = f

0(x, v) ,

−∇x

φǫ = Eǫ , −∆x

φǫ =

∫

R2

fǫ dv ,

Ξǫ(t, x) = −H

0

ǫx+ H

1

(

ω1

t

ǫ

)

x .

(t plays the role of longitudinal oordinate)






Numeri al results


Polar oordinates

Polar oordinates :

x = r os θ , v

r

= v

x

os θ + v

y

sin θ ,y = r sin θ , vθ = v

y

os θ − v

x

sin θ .

Conservation of the angular momentum rvθ along the traje tories,

Null angular momentum at the beam sour e,

∂t

fǫ +v

r

ǫ∂r

fǫ + (Eǫ + Ξǫ) ∂vr

fǫ = 0 ,

fǫ(t = 0, r , vr

) = f

0(r , vr

) ,1

r

∂r

(r Eǫ) =

∫

R

f

ǫdv

r

,

Ξǫ(t, r) = −1

ǫH

0

r + H

1

(

ω1

t

ǫ

)

r .






Numeri al results


0-th order Two-S ale model

Assumptions :

H

0

= 1,

f

0

∈ L

1

(

R2; |r | dr dvr

)

∩ L

p

(

R2; |r | dr dvr

)

(p ≥ 2),

f

0

(r , vr

) ≥ 0 for all (r , vr

) ∈ R2

,

∫

R2

(

r

2 + v

2

r

) f0

(r , vr

) |r | dr dvr

< +∞.






Numeri al results


0-th order Two-S ale model

Theorem (Frénod, Salvarani & Sonnendrü ker - 2009)

fǫ → F = F (t, τ, r , vr

) ∈ L

∞(

0,T ; L∞#(

0, 2π; Lp(R2; |r | dr dvr

))

)

two-s ale,

Eǫ → E = E(t, τ, r) ∈ L

∞(

0,T ; L∞#

(

0, 2π;W 1,3/2(R; |r | dr)))

two-s ale.

Moreover, ∃G = G(t, q, ur

) ∈ L

∞(

0,T ; Lp(R2; |r | dr dvr

))

su h that

F (t, τ, r , vr

) = G (t, os(τ) r − sin(τ) vr

, sin(τ) r + os(τ) vr

) ,

with

∂t

G(t, q, ur

) +

[

∫

2π

0

(

− sin(σ) os(σ)

)

[

Er

(

t, σ, os(σ) q + sin(σ) ur

)

+IQ(ω1

)

2πH

1

(ω1

σ)(

os(σ) q + sin(σ) ur

)

]

dσ

]

· ∇(q,ur

)G(t, q, ur

) = 0 ,

G(t = 0, q, ur

) =1

2πf

0(q, ur

) ,

1

r

∂r

(

r Er

(t, τ, r))

=

∫

R

G

(

t, os(τ) r − sin(τ) vr

, sin(τ) r + os(τ) vr

)

dv

r

,






Numeri al results


1


Introdu tion




2




Numeri al results







Numeri al results


Prin iple of Ba kward Semi-Lagrangian (BSL) method

Consider the onservative transport equation given by

∂t

f (t, x) +U(t, x) · ∇x

f (t, x) = 0 ,

and asso iated hara teristi s t 7→ X(t; x; s) whi h are solutions of

∂t

X(t) = U (t,X(t)) ,X(s) = x .

=⇒ f is onserved along the hara teristi s :

f (t,X(t; x; s)) = f (s, x) .






Numeri al results


Prin iple of BSL method

PSfrag repla ements

X(t; xi

; tn+1)

x

i

t

n

t

n+1

Prin iple :

f (tn+1, xi

) = f

(

t

n,X(tn; xi

; tn+1))

.

In most ases, f is only known on

mesh nodes (xi

)i

so we repla e the

onservation equation by

f (tn+1, xi

) ≈ Πf(

t

n,X(tn; xi

; tn+1))

,

where Π is an interpolation operator

based on the points (xi

)i

.






Numeri al results


Prin iple of BSL method

Denoting f

n

the approximation of f (tn, ·), an iteration of the BSL method

writes :

1 Compute X(tn; xi

; tn+1) by solving

∂t

X(t) = U (t,X(t)) ,X(tn+1) = x

i

,

2 Assuming that f

n(xk

) is known for all k, we ompute f

n+1(xi

) as follows :

f

n+1(xi

) = Πf n(

X(tn; xi

; tn+1))

.

Resolution of the hara teristi s (Sonnendrü ker et al. - 1999)

Use of a se ond order Taylor expansion :

X(tn−1; xi

; tn+1) ≈ x

i

− 2d

n(xi

) ,

with d

n(xi

) = ∆t (I+∆t∇x

(ΠU)(tn, xi

))−1

U(tn, xi

).






Numeri al results


Meshes and rotation of the beam

In a Two-S ale BSL method, assuming Supp(f 0) ⊂ Ω = [−R,R]× [−v

R

, vR

]does not ne essarly implies that the support of

(r , vr

) 7−→ f

0

(

os(τ ) r − sin(τ ) vr

, sin(τ ) r + os(τ ) vr

)

,

is in luded in Ω.

PSfrag repla ements

R

v

R

−R

−v

R

−R − v

R

R + v

R

v

R

+ R

−v

R

− R

r

v

r






Numeri al results


Extended mesh

First solution : Extend Ω into Ω′in order to guarantee that Supp (G(t, ·, ·)) ⊂ Ω′

for

all t.

=⇒ If Supp(f 0) ⊂ Ω = [−R,R] × [−v

R

, vR

], G should be approa hed on

Ω′ = [−Q

m

,Qm

]× [−U

m

,Um

] with

Q

m

≥ R + v

R

U

m

≥ R + v

R

.

PSfrag repla ements

R

v

R

−R

−v

R

−R − v

R

R + v

R

v

R

+ R

−v

R

− R

r

v

r






Numeri al results


Extended mesh

Dene the extended domain Ω′ = [−Q

m

,Qm

]× [Um

,Um

], and the uniform

mesh M(Ω′) as

M(Ω′) =

(qi

, ur

j

) = (i∆q, j ∆u

r

) : i = −P

q

, . . . ,Pq

, j = −P

u

r

, . . . ,Pu

r

,

Dene on [0, 2π] the following uniform mesh :

M

(

[0, 2π])

=

τm

= m∆τ : m = 0, . . . ,Pτ

.

Dene the ubi spline interpolation operators Π1

on [−Q

m

,Qm

] and Π2

on Ω′,

Dene G

n

as the approximation of G (tn, ·, ·), and En(τ, ·) as theapproximation of E(tn, τ, ·),Dene R as

R : R2 × [0, 2π] −→ R2

(r , vr

, τ ) 7−→(

os(τ ) r − sin(τ ) vr

, sin(τ ) r + os(τ ) vr

) ,






Numeri al results


Extended mesh : sket h of the BSL method

Assume that G

n−1

and G

n

are known

1 Compute En

:

En(τm

, qi

) =

1

q

i

∫

q

i

0

∫

U

m

−U

m

s Π2

G

n (R(τm

, s, vr

)) ds dvr

, if i > 0,

0 , else,

with En(τm

,−q

i

) = −En(τm

, qi

),

2 Compute 〈En〉(qi

, ur

j

) dened as

〈En〉(qi

, ur

j

) =

∫

2π

0

(

− sinσ os σ

)

[

Π1

En (σ, os(σ) qi

+ sin(σ) ur

j

)

+IQ(ω1

)

2πH

1

(ω1

σ) ( os(σ) qi

+ sin(σ) ur

j

)]

dσ ,






Numeri al results


Extended mesh : sket h of the BSL method

3 Compute d

n

i,j :

d

n

i,j = ∆t (An

i,j )−1 〈En〉(q

i

, ur

j

) ,

with An

i,j dened as

An

i,j = I+∆t

(

∇(q,ur

)Π2

〈En〉)

(qi

, ur

j

) .

4 Compute G

n+1:

G

n+1(qi

, ur

j

) = Π2

G

n−1

((

q

i

u

r

j

)

− 2d

n

i,j

)

.

5 Compute the approximation of fǫ(tn+1, ·, ·) given by

fǫ(tn+1, r , v

r

) ≈ 2πΠ2

G

n+1

(

R

(

t

ǫ, r , v

r

))

.






Numeri al results


Rotating meshes

Se ond solution : Take into a ount the rotations within the two-s ale model.

For all τ ∈ [0, 2π], dene Ω(τ ) and M

(

Ω(τ ))

as

Ω(τ ) = R(τ,Ω) , M((Ω(τ )) = R (τ,M(Ω)) ,

with

Ω = [−R,R]× [−v

R

, vR

] ,

M(Ω) = (ri

, vr

j

) = (i ∆r , j ∆v

r

) : i = −P

r

, . . . ,Pr

, j = −P

v

r

, . . . ,Pv

r

,

M ([−R,R]) = ri

= i ∆r : i = −P

r

, . . . ,Pr

,

We also dene the uniform mesh M ([0, 2π]) as

M ([0, 2π]) = τm

= m∆τ : m = 0, . . . ,Pτ .






Numeri al results


Rotating meshes

Prin iple

Approa h E(tn, ·, ·) on M ([−R,R])×M ([0, 2π]),

Approa h F (tn, ·, ·, ·) on M(Ω)×M ([0, 2π]),

Approa h G (tn, ·, ·) on M (Ω(τm

)) for ea h τm

∈ M ([0, 2π]).

Lang et al. - 2003.

Consequen es :

For all f : R2 → R with ompa t support in luded in Ω, we have

Supp(f ) ⊂ Ω ⇐⇒ Supp (f R(τ, ·, ·)) ⊂ Ω(τ ) for all τ ∈ [0, 2π],

We have F (t, τ, r , vr

) = G (t,R(τ, r , vr

)).






Numeri al results


Rotating meshes

PSfrag repla ements

r

v

r

PSfrag repla ements

r

v

r

r

v

r

Figure: Mesh M

(

Ω(τ))

and support of f

0 R(τ, ·, ·) for τ = 0 (left) and τ = π3

(right).






Numeri al results


Rotating meshes

G

n

is the approximation of G (tn, ·, ·) on the meshes M (Ω(τm

)),

En

is the approximation of E(tn, ·, ·) the mesh M ([−R,R])×M ([0, 2π]),

t

n = n∆t with ∆t dened as

∆t = ǫK ∆τ , K ∈ N∗xed,

Π1

is a 1D ubi spline interpolation operator based on M ([−R,R]),

For all m = 0, . . . ,Pτ , Π2,m is an interpolation operator su h that, for all

g : Ω(τm

) → R,

Π2,mg : Ω(τ

m

) −→ R

(q, ur

) 7−→ Π2,mg(q, ur ) = Π

2,0g (R(−τm

, q, ur

))

where Π2,0 = Π

2

is a ubi spline interpolation operator based on the

mesh M(Ω).






Numeri al results


Rotating meshes

Assume that G

n

and G

n−1

are known on ea h M (Ω(τm

)) :

1 Compute En(τm

, ri

) :

En(τ, ri

) = −En(τm

,−r

i

) =

1

r

i

∫

r

i

0

∫

v

R

v

R

s G

n R(τm

, s, vr

) dvr

ds , if i > 0,

0 , else,

2 Comute 〈En〉 on ea h M

(

Ω(τm

))

:

(〈En〉 R) (τm

, ri

, vr

j

)

=

∫

2π

0

(

− sin(σ) os σ

)

[

Π1

En(

σ, os(σ − τm

) ri

+ sin(σ − τm

) vr

j

)

+IQ(ω1

)

2πH

1

(ω1

σ)(

os(σ − τm

) ri

+ sin(σ − τm

) vr

j

)

]

dσ ,






Numeri al results


Rotating meshes

3 Compute d

n

m,i,j dened as

d

n

m,i,j = ∆t (An

m,i,j )−1 〈(En〉 R)(τ

m

, ri

, vr

j

) ,

with An

m,i,j dened by

An

m,i,j = I+∆t∇(q,ur

) (Π2,m (〈En〉 R)) (τm

, ri

, vr

j

) ,

4 Compute G

n+1on ea h mesh M (Ω(τ

m

)) :

G

n+1 R(τm

, ri

, vr

j

) = Πm

2

G

n−1

(

R(τm

, ri

, vr

j

)− 2 d

n

m,i,j

)

,

5 Compute the approximation of fǫ(tn+1, ·, ·) on M(Ω) given by

fǫ(tn+1, r

i

, vr

j

) ≈ 2π G

n+1 R(τ(n+1)K , ri , vr j ) .






Numeri al results


Extended mesh VS Rotating meshes

Extended mesh :

Need to rene the mesh in q and u

r

dire tions,

Need additional interpolations for omputing En

and re ontru t and

approximation of fǫ,

The implementation is quite simple,

G is dis retized on (2Pq

+ 1)× (2Pu

r

+ 1) points,

Rotating meshes :

We do not need to rene the mesh in r and v

r

dire tions,

No additional interpolation for omputing En

the approximation of fǫ,

Take into a ount hanges of variables,

G is dis retized on (2Pr

+ 1)× (2Pv

r

+ 1)× (Pτ + 1) points.






Numeri al results


1


Introdu tion




2




Numeri al results







Numeri al results


Classi al BSL method

Apply a Ba kward Semi-Lagrangian method on the singularly perturbed

Vlasov-Poisson model :

∂t

fǫ +v

r

ǫ∂r


fǫ = 0 ,

fǫ(t = 0, r , vr

) = f

0(r , vr

) ,1

r

∂r

(r Eǫ) =

∫

R

fǫ dvr ,

Ξǫ(t, r) = −1

ǫH

0

r + H

1

(

ω1

t

ǫ

)

r .

Strang's splitting =⇒ We repla e the resolution of Vlasov equation by the

su essive resolution of

∂t

fǫ +v

r

ǫ∂r

fǫ = 0 ,

and

∂t


fǫ = 0 .






Numeri al results



We take Ω = [−R,R]× [vR

, vR

] and the uniform mesh M(Ω) dened by

M(Ω) = (ri

, vr

j

) = (i∆r , j ∆v

r

) : i = −P

r

, . . . ,Pr

, j = P

v

r

, . . . ,Pv

r

,

Dene the 1D ubi spline interpolation operators Πr

and Πv

r

,

Dene f

n

ǫ as the approximation of fǫ(tn, ·, ·), et E n

ǫ (tn, ·) as the

approximation of Eǫ(tn, ·).






Numeri al results



Assume that f

n

ǫ (ri

, vr

j

) and E

n

ǫ (ri ) are known :

1 Half adve tion in v

r

:

f

∗ǫ (r

i

, vr

j

) = Πv

r

f

n

ǫ

(

r

i

, vr

j

−∆t

2

E

n

ǫ (ri )−∫

t

n+1/2

t

n

Ξǫ(θ, ri

) dθ

)

,

2 Adve tion in r :

f

∗∗ǫ (r

i

, vr

j

) = Πr

f

∗ǫ

(

r

i

−∆t

ǫv

r

j

, vr

j

)

,

3 ompute E

n+1ǫ :

E

n+1ǫ (r

i

) = −E

n+1ǫ (−r

i

) =

1

r

i

∫

r

i

0

∫

v

R

−v

R

s f

∗∗ǫ (s, v

r

) dvr

ds , if i > 0,

0 , else,

4 Half-adve tion in v

r

:

f

n+1ǫ (r

i

, vr

j

) = Πv

r

f

∗∗ǫ

(

r

i

, vr

j

−∆t

2

E

n+1ǫ (r

i

)−∫

t

n+1

t

n+1/2

Ξǫ(θ, ri

) dθ

)

.






Numeri al results



Problem : In the adve tion in r , we have have r

i

− ∆t

ǫv

r

j

/∈ [−R,R] even if

(ri

, vr

j

) ∈ Ω when ǫ is small :

=⇒ f

∗∗ǫ (r

i

, vr

j

) = Πr

f

∗ǫ

(

r

i

−∆t

ǫv

r

j

, vr

j

)

= 0 .

Same problem within the half-adve tions in v

r

due to the denition of Ξǫ.

=⇒ When ǫ is small, a very small ∆t is required.

Take ∆t satisfying

r

i

−∆r ≤ r

i

−∆t

ǫv

r

j

≤ r

i

+∆r ,

v

r

j

−∆v

r

≤ v

r

j

−∆t

2

E

n

ǫ (ri )−∫

t

n+1/2

t

n

Ξǫ(θ, ri

) dθ ≤ v

r

j

+∆v

r

,

v

r

j

−∆v

r

≤ v

r

j

−∆t

2

E

n+1ǫ (r

i

)−∫

t

n+1

t

n+1/2

Ξǫ(θ, ri

) dθ ≤ v

r

j

+∆v

r

.






Numeri al results



No "arti ial" loss of data but

∆t = O(ǫ),

The number of time iterations required for rea hing the nal time T will

blow up when ǫ is small,

The total number of interpolations in r and v

r

will also be in reased =⇒Need to rene the mesh M(Ω) for ontrolling these errors.






Numeri al results


Linear ases

Negle t the self- onsistent ele tri eld =⇒ The two-s ale limit model redu es

itself to

∂t

G(t, q, ur

)

+

[

IQ(ω1

)

2π

∫

2π

0

(

− sin(σ) os(σ)

)

H

1

(ω1

σ) ( os(σ) q + sin(σ) ur

) dσ

]

·∇(q,ur

)G(t, q, ur

) = 0 ,

G(t = 0, q, ur

) =1

2πf

0

(q, ur

) .

Up to a good hoi e of H

1

and ω1

, we an exhibit the analyti formulation of

G .






Numeri al results


Linear ases

Four types of simulation are performed :

Type (I) : Classi al BSL method on Ω and M(Ω) hara terized with

R = v

R

= 3 and P

r

= P

v

r

= 64,

Type (II) : Classi al BSL method on Ω and M(Ω) hara terized with

R = v

R

= 3 and P

r

= P

v

r

= 128,

Type (III) : Two-S ale method on rotating meshes with Ω and M(Ω) hara terized with R = v

R

= 3, P

r

= P

v

r

= 64 and Pτ = 16,

Type (IV) : Two-S ale method on extended mesh with Ω′, Ω, M(Ω′) and

M(Ω) hara terized with R = v

R

= 3, Q

m

= U

m

= R + v

R

= 6,

P

q

= P

u

r

= 128 and Pτ = 16, where the approximation of f

ǫis

re onstru ted on M(Ω) with P

r

= P

v

r

= 64.

Semi-Gaussian initial distribution :

f

0(r , vr

) =n

0

√2π v

th

exp

(

− v

2

r

2 v

2

th

)

I[−r

m

,rm

](r) ,

where r

m

= 0.75, vth

= 0.1, and n

0 = 4.






Numeri al results


Non-resonant ase

Assume that ω1

/∈ Q. Then the transport equation for G is redu ed to

∂t

G = 0 ,

so G writes as

G (t, q, ur

) =1

2πf

0

(q, ur

) , ∀ (t, q, ur

) ∈ [0,T ]× R2 .

Then, the approximation of fǫ we rebuild is the following fun tion :

(t, r , vr

) 7−→ f

0

(

os

(

t

ǫ

)

r − sin

(

t

ǫ

)

v

r

, sin

(

t

ǫ

)

r + os

(

t

ǫ

)

v

r

)

.






Numeri al results


Non-resonant ase

Numeri al results at time t = 1.1088 with ω1

= 4

√2, H

1

(τ ) = os(τ ) andǫ = 10

−2

:

(I) (II)

(III) (IV)






Numeri al results


Non-resonant ase


= 4

√2, H

1

(τ ) = os(τ ) andǫ = 10

−2

:

(I) (II)

(III) (IV)






Numeri al results


Non-resonant ase

Evolution of the error in L

1

norm between

(t, r , vr

) 7−→ f

0

(

os

(

t

ǫ

)

r − sin

(

t

ǫ

)

v

r

, sin

(

t

ǫ

)

r + os

(

t

ǫ

)

v

r

)

,

and the approximations obtained from simulations (I), (II), (III) and (IV).

0 1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

time

(I)

(II)

(III)

(IV)






Numeri al results


Resonant ase

Assume that ω1

∈ N≥ 2

and H

1

(τ ) = os

2(τ ). Then the transport equation for

G is redu ed to

∂t

G −u

r

4

∂q

G +q

4

∂u

r

G = 0 ,

G (t = 0, q, ur

) =1

2πf

0(q, ur

) ,

and the analyti expression of G is

G (t, q, ur

) =1

2πf

0

(

os

(

t

4

)

q − sin

(

t

4

)

u

r

, sin

(

t

4

)

q + os

(

t

4

)

u

r

)

.

=⇒ f

ǫis approa hed by the following fun tion

(t, r , vr

) 7−→ f

0

(

os

(

t

ǫ+ t

4

)

r − sin

(

t

ǫ+ t

4

)

v

r

, sin(

t

ǫ+ t

4

)

r + os

(

t

ǫ+ t

4

)

v

r

)

.






Numeri al results


Resonant ase


= 2, H

1

(τ ) = os

2(τ ) andǫ = 10

−2

:

(I) (II)

(III) (IV)






Numeri al results


Non-linear ases

The self- onsistent ele tri is no more negle ted :

=⇒ Find an analyti expression of G is a big hallenge !

=⇒ Validate the Two-S ale numeri al method by omparing it to the lassi al

method.

Semi-Gaussian initial distribution :

f

0(r , vr

) =n

0

√2π v

th

exp

(

− v

2

r

2 v

2

th

)

I[−r

m

,rm

](r) ,

with r

m

= 0.75, vth

= 0.1, and n

0 = 4.






Numeri al results


Non-linear ases

Four types of simulations :


R = v

R

= 3 and P

r

= P

v

r

= 64,

Type (II) : Classi al BSL method on Ω and M(Ω) hara terized with

R = v

R

= 3 and P

r

= P

v

r

= 256,

Type (III) : Two-S ale method on rotating meshes with Ω and M(Ω) hara terized with R = v

R

= 3, P

r

= P

v

r

= 64 and Pτ = 16,

Type (IV) : Two-S ale method on extended mesh with Ω′, Ω, M(Ω′) and


R

= 3, Q

m

= U

m

= R + v

R

= 6,

P

q

= P

u

r


ǫis


r

= P

v

r

= 64.






Numeri al results


Non-resonant ase

Assume that ω1

/∈ Q.

Assume that the time step for Two-S ale simulations (III) and (IV) is of

the form

∆t

H

= ǫK ∆τ ,

with K = 5 (in pra ti e, we take ∆t

H

≈ 0.0185),

Assume that the time step for the lassi al simulations (I) and (II) is of

the form

∆t

NH

=∆t

H

N

,

with N large enough for insuring the stability of the s heme.






Numeri al results


Non-resonant ase


= 4

√2, H

1

(τ ) = os(τ ) andǫ = 10

−2

:

(I) (II)

(III) (IV)






Numeri al results


Resonant ase

Assume that ω1

∈ N≥ 2

.

Assume that the time step used for Two-S ale simulations (III) and (IV) is

of the form

∆t

H

= ǫK ∆τ ,

with K = 2 (in pra ti e, we take ∆t

H

≈ 7.392 × 10

−3

),

Assume that the time step for lassi al simulations (I) and (II) is of the

form

∆t

NH

=∆t

H

N

,

with N large enough for insuring that the method is stable.






Numeri al results


Resonant ase


= 2, H

1

(τ ) = os

2(τ ) andǫ = 10

−2

:

(I) (II)

(III) (IV)






Numeri al results


CPU time ost

Test (I) (II) (III) (IV)

CPU time N CPU time N CPU time CPU time

ω1

= 4

√2

H

1

= os

35m 122 35h 6m 50s 480 1h 43m 39s 55m 3s

ω1

= 2

H

1

= os

2

37m 32s 49 38h 7m 6s 192 5h 45m 25s 2h 37m 25s

Table: The nal time is T = 6.93 for the non-resonant ase, and T = 6.9854 for the

resonant ase.






Numeri al results


A last test ase

Consider the following semi-gaussian initial distribution :

f

0(r , vr

) =n

0

√2π v

th

exp

(

− v

2

r

2 v

2

th

)

I[−r

m

,rm

](r) ,

with r

m

= 1.85, vth

= 0.1 and n

0 = 4.






Numeri al results


A last test ase

3 types of simulations are onsidered :


R = v

R

= 3 and P

r

= P

v

r

= 256,

Type (II) : Two-S ale method on rotating meshes with Ω and M(Ω) hara terized with R = v

R

= 3, P

r

= P

v

r

= 128 and Pτ = 20,

Type (III) : Two-S ale method on extended mesh with Ω′, Ω, M(Ω′) and


R

= 3, Q

m

= U

m

= R + v

R

= 6,

P

q

= P

u

r


ǫis


r

= P

v

r

= 128.






Numeri al results


A last test ase


= 1, H

1

(τ ) = os

2(τ ) andǫ = 10

−2

:

(I) (II) (III)






Numeri al results


Con lusions

A 0-th order Two-S ale numeri al s heme has been developed for a

Vlasov-Poisson model : it is based on a Ba kward Semi-Lagrangian

pro edure and the use of rotating meshes,

The use of rotating meshes allows to redu e the total number of

interpolations,

Both Two-S ale methods do not require a mesh in (r , vr

) as rened as it is

needed for running a lassi al BSL method for apturing the lamentation

phenomena :

Smaller numeri al diusion introdu ed within long time simulations,

Two-S ale methods are mu h faster than the lassi al BSL method, even if

we onsider rotating meshes,

In pra ti e, the use of rotating meshes ae t the CPU time ost of the

Two-S ale method, however it redu es drasti ally the error of the s heme

on linear ases,

The numeri al results from Frénod, Salvarani & Sonnendrü ker (2009) are

onrmed with another type of Two-S ale numeri al method.






Numeri al results


Perspe tives

Apply su h a method on other Vlasov-type problems,

Consider other external ele tri and magneti elds with high amplitude

and/or high frequen y os illations,

Repla e BSL method by FSL within the Two-S ale method to ta kle the

xed point problem,

First order Two-S ale s hemes.


Asymptotic-Preserving Schemes VS Two-Scale Schemes - Part II

Documents

Transcript of Asymptotic-Preserving Schemes VS Two-Scale Schemes - Part II