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
High order TS expansion
0-th order TS-BSL numeri al s heme
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,
...
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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,
...
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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,
...
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
0-th order Two-S ale expansion
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
Chara terization of U
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
)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
)
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
High Order Two-S ale expansion
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
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.
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
?
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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)) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
Chara terization of U
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
))
,
is bounded independently of ǫ in(
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
Chara terization of U
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
∞ (0,T ; Lp(Rn)) is the solution of
∂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)) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
First 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 W
1,h from V
0,h,
4 Apply a lassi al numeri al method for solving (TS
1
) =⇒ nd
V
1,h = V
1,h(t, x),
5 Compute U
1,h(t, τ, x) by dis retizing (RB
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
).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
High Order Two-S ale expansion
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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)) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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))
)
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
Chara terization of U
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
Chara terization of U
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
∞ (0,T ; Lp(Rn)) is the solution of
∂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)] .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
N-th order Two-S ale S heme
For all i = 0, . . . ,N, we perform the following omputations :
1 Apply a lassi al numeri al method for solving (TS
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
).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Introdu tion
0-th order Two-S ale expansion
High Order Two-S ale expansion
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Paraxial approximation
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Paraxial approximation
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
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
< +∞.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
,
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
) ,
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 ,
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
))
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
)).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 ,
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 ) .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
Con lusions and perspe tives
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 + (Eǫ + Ξǫ) ∂vr
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 + (Eǫ + Ξǫ) ∂vr
fǫ = 0 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Classi al BSL method
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, ·).
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Classi al BSL method
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
)
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Classi al BSL method
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
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Classi al BSL method
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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 .
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
)
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Non-resonant ase
Numeri al results at time t = 6.468 with ω1
= 4
√2, H
1
(τ ) = os(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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
)
.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Resonant ase
Numeri al results at time t = 0.2957 with ω1
= 2, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Resonant ase
Numeri al results at time t = 5.9875 with ω1
= 2, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Non-linear ases
Four types of simulations :
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
= 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
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Non-resonant ase
Numeri al results at time t = 1.4784 with ω1
= 4
√2, H
1
(τ ) = os(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Non-resonant ase
Numeri al results at time t = 3.234 with ω1
= 4
√2, H
1
(τ ) = os(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Non-resonant ase
Numeri al results at time t = 5.544 with ω1
= 4
√2, H
1
(τ ) = os(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Resonant ase
Numeri al results at time t = 1.1458 with ω1
= 2, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Resonant ase
Numeri al results at time t = 3.6221 with ω1
= 2, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
Resonant ase
Numeri al results at time t = 5.8027 with ω1
= 2, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II)
(III) (IV)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
A last test ase
3 types of simulations are onsidered :
Type (I) : Classi al BSL method on Ω and M(Ω) hara terized with
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
M(Ω) hara terized with R = v
R
= 3, Q
m
= U
m
= R + v
R
= 6,
P
q
= P
u
r
= 256 and Pτ = 20, where the approximation of f
ǫis
re onstru ted on M(Ω) with P
r
= P
v
r
= 128.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
A last test ase
Numeri al results at time t = 1.3464 with ω1
= 1, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II) (III)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
A last test ase
Numeri al results at time t = 4.3388 with ω1
= 1, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II) (III)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
A last test ase
Numeri al results at time t = 5.1462 with ω1
= 1, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II) (III)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
A last test ase
Numeri al results at time t = 5.984 with ω1
= 1, H
1
(τ ) = os
2(τ ) andǫ = 10
−2
:
(I) (II) (III)
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II
High order TS expansion
0-th order TS-BSL numeri al s heme
Two-S ale modelization
Semi-Lagrangian method
Numeri al results
Con lusions and perspe tives
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.
Alexandre MOUTON (CNRS - Lille) Asymptoti -Preserving S hemes VS Two-S ale S hemes - Part II