Congrès SMAI 2019

Convergence Analysis of Asymptotic Preserving Schemes
for strongly magnetized plasmas
Hamed Zakerzadeh¶
Institut de Math´ematiques de Toulouse, Universit´e Toulouse III - Paul Sabatier
joint work with: F. Filbet (Toulouse), M. Rodrigues (Rennes)
SMAI 2019
May 16 th 2019
¶
supported by LabEx CIMI Toulouse and Institut Universitaire de France

Introduction Continuous estimates Discrete estimates Numerical example References
Outline
Introduction
Continuous estimates
Discrete estimates
Numerical example
1/19

Fusion reactor (TOKAMAK):
Тороидальная Камера с Магнитными Катушками
“toroidal chamber with magnetic coils”
requires a quite large magnetic ﬁeld!
very challenging to control the plasma in the core!
not economical yet!
2/19

Fusion reactor (TOKAMAK):
Тороидальная Камера с Магнитными Катушками
“toroidal chamber with magnetic coils”
requires a quite large magnetic ﬁeld!
very challenging to control the plasma in the core!
not economical yet!
Vlasov equation: evolution of charged particles



∂f
∂t
+ divx (vf ) + divv (Ff ) = 0
f (0, ·, ·) = f0
electro-magnetic force ﬁeld: F =
q
m
(E + v × B)
2/19

Vlasov systems:
Vlasov-Maxwell system
∂f
∂t
+ divx (vf ) + divv ((E + v × B) f ) = 0, (x, v) ∈ R4
,
where E and B are solutions of Maxwell equations:



∂t E − c2 × B = − J
0
,
∂t B + × E = 0,
· E =
0
, · B = 0
with
(t, x) := q f (t, x, v)dv, J(t, x) := q vf (t, x, v)dv.
3/19

Long-time Vlasov–Poisson system
with an external uniform magnetic ﬁeld B(t, x) = 1
ε
(0, 0, 1)T :
ε
∂f ε
∂t
+ divx (vf ε
) + divv (E −
1
ε
v⊥
)f ε
= 0 ,
where E = − x φ such that −∆x φ = ε0
.
Singular limit ε → 0:
limit system for the weak limit f ε f :
∂f
∂t
− E⊥
· x f −
1
2
∆φ v⊥
· v f = 0 , (x, v) ∈ R4
,
limit of charge density:
∂
∂t
− x · E⊥
= 0 , x ∈ R2
.
4/19

Particle-In-Cell (PIC) methods:
(i) Solve the characteristic system for N macro-particles:



εxε(t) = vε(t), xε(t0) = x0
ε ,
εvε(t) = E(t, xε(t)) −
1
ε
v⊥
ε (t), vε(t0) = v0
ε .
(ii) approximate f ε:
fN,α(t, x, v) :=
1≤k≤N
ωk ϕα(x − xk (t)) ϕα(x − vk (t)).
Proposition [Cohen and Perthame, 2000]
lim
α→0
lim
N→∞
f (t) − fN,α(t) Lp → 0, 1 ≤ p ≤ ∞,
with N number of particles and ϕα as a smooth approximation of the Dirac mass s.t.:
numerical noise
more eﬃcient compared to direct methods (in phase space)
5/19

Stiﬀness!
very stiﬀ system of ODEs where ε 1:



εxε(t) = vε
εvε(t) = E(t, xε) −
1
ε
v⊥
ε
ε→0
−−−→ lim
ε→0
x(t) =?
6/19

Stiﬀness!



εxε(t) = vε
1
ε
v⊥
ε
ε→0
−−−→ lim
ε→0
x(t) =?
Guiding center approximation
E-cross-B drift: slow drift normal to E, in a plane perpendicular to B:
x (t) = −E⊥
(t, x(t))
(Wikipedia)
6/19

Stiﬀness!



εxε(t) = vε
1
ε
v⊥
ε
ε→0
−−−→ lim
ε→0
x(t) =?
Guiding center approximation
E-cross-B drift: slow drift normal to E, in a plane perpendicular to B:
x (t) = −E⊥
(t, x(t))
(Wikipedia)
[Filbet and Rodrigues, 2017]
highly oscillatory ∼ ε−2 → capture macroscopic behaviour with ∆t = O(1)
6/19

Asymptotic Preserving (AP) schemes
Introduced by [Jin, 1999]
- [Il’in, 1969]: BVP
- [Larsen et al., 1987] : Neutron transport
asymptotic eﬃciency: uniform eﬃciency wrt ε
asymptotic consistency: consistent with the asymptotic system as ε → 0
asymptotic stability: uniformly stable in ε
7/19

Uniform convergence
Estimate the error Uε
∆ − Uε for an r-th order scheme:
E2 = O(∆r
/ε)
8/19

Uniform convergence
[Jin, 2010]
E1 = O(∆r
+ ε)
E2 = O(∆r
/ε)
Uniform estimate:
Uε
∆ − Uε
≤ min(E1, E2).
8/19

Uniform convergence
[Jin, 2010]
E1 = O(∆r
+ ε)
E2 = O(∆r
/ε)
Uniform estimate:
Uε
∆ − Uε
≤ min(E1, E2).
Can we improve E1 and E2?
E1 = O(∆r
+ εq
), q > 1,
E2 = O(
∆r
εp
), p < 1.
Uε
∆ − Uε ≈ (∆r )
q
p+q
8/19

Outline
Introduction
Discrete estimates
Numerical example
8/19

Oscillatory limit of the continuous model
yε(t) := xε(t) − ε v⊥
ε (t) slower than xε: yε(t) = −E⊥
(t, xε(t))
(Xε(t, s, xs
ε, vs
ε), Vε(t, s, xs
ε, vs
ε), Yε(t, s, xs
ε, vs
ε)) := (xε(t), vε(t), yε(t))
K0 := E L∞ , Kt := ∂t E L∞ , Kx := dx E L∞ , Kxx := d2
x E L∞ .
9/19

Oscillatory limit of the continuous model
yε(t) := xε(t) − ε v⊥
ε (t) slower than xε: yε(t) = −E⊥
(t, xε(t))
(Xε(t, s, xs
ε, vs
ε), Vε(t, s, xs
ε, vs
ε), Yε(t, s, xs
ε, vs
ε)) := (xε(t), vε(t), yε(t))
K0 := E L∞ , Kt := ∂t E L∞ , Kx := dx E L∞ , Kxx := d2
x E L∞ .
Theorem [Filbet et al., 2019]
(i) Assume that E ∈ W 1,∞. Then, for any ε > 0:
Xε(t, 0, x0
ε , v0
ε ) − X(t, 0, x0
ε )
E
ε eKx t
(1 + t2
) ( v0
ε + ε) .
(ii) Assume that E ∈ W 2,∞. Then, for any ε > 0:
Yε(t, 0, x0
ε , v0
ε ) − X(t, 0, x0
ε − ε(v0
ε )⊥
)
E
ε2
(1 + t4
) e2 Kx t
1 + ε2
+ v0
ε
2
.
9/19

Proof
1. zε(t) := vε + ε E⊥
such that zε(t) = −
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
10/19

Proof
1. zε(t) := vε + ε E⊥
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
2. zε(t) ≤ eKx t v0
ε + ε K0 + ε t eKx t (Kt + Kx K0)
10/19

Proof
1. zε(t) := vε + ε E⊥
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
3. vε(t) ≤ zε(t) + K0 ε
10/19

Proof
1. zε(t) := vε + ε E⊥
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
3. vε(t) ≤ zε(t) + K0 ε
4. Taylor expansion:
yε(t) = −E⊥
(t, yε(t) + εv⊥
ε (t))
= −E⊥
(t, yε) − ε dx E⊥
(t, yε) v⊥
ε + ε2
Θ0(t, yε, vε) ,
with the remainder Θ0 such that Θ0(t, yε, vε) ≤ 1
2
Kxx vε
2 .
10/19

Proof
1. zε(t) := vε + ε E⊥
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
3. vε(t) ≤ zε(t) + K0 ε
yε(t) = −E⊥
(t, yε(t) + εv⊥
ε (t))
= −E⊥
(t, yε) v⊥
ε + ε2
Θ0(t, yε, vε) ,
2
Kxx vε
2 .
5. employ εvε(t) = E(t, xε) −
1
ε
v⊥
ε
yε − ε3
dx E⊥
(t, yε)vε = − E⊥
(t, yε) − ε2
dx E⊥
(t, yε)E(t, xε) + Θ0(t, yε, vε)
− ε3
∂t dx E⊥
(t, yε) − d2
x E⊥
(t, yε)E⊥
(t, xε) vε ,
10/19

Proof
1. zε(t) := vε + ε E⊥
1
ε2
z⊥
ε (t) + ε
d
dt
E⊥
(t, xε(t))
3. vε(t) ≤ zε(t) + K0 ε
yε(t) = −E⊥
(t, yε(t) + εv⊥
ε (t))
= −E⊥
(t, yε) v⊥
ε + ε2
Θ0(t, yε, vε) ,
2
Kxx vε
2 .
5. employ εvε(t) = E(t, xε) −
1
ε
v⊥
ε
yε − ε3
dx E⊥
(t, yε)vε = − E⊥
(t, yε) − ε2
dx E⊥
(t, yε)E(t, xε) + Θ0(t, yε, vε)
− ε3
∂t dx E⊥
(t, yε) − d2
x E⊥
(t, yε)E⊥
(t, xε) vε ,
6. ﬁnd the diﬀerence with x (t) = −E⊥(t, x(t)) and integrate.
10/19

Outline
Introduction
Discrete estimates
Numerical example
10/19

First-order IMEX scheme



ε
xn+1
ε − xn
ε
∆t
= vn+1
ε
ε
vn+1
ε − vn
ε
∆t
= E(tn, xn
ε ) −
1
ε
(vn+1
ε )⊥
ε→0
−−−→
xn+1 − xn
∆t
= −E⊥
(tn
, xn
)
11/19

First-order IMEX scheme



ε
xn+1
ε − xn
ε
∆t
= vn+1
ε
ε
vn+1
ε − vn
ε
∆t
= E(tn, xn
ε ) −
1
ε
(vn+1
ε )⊥
ε→0
−−−→
xn+1 − xn
∆t
= −E⊥
(tn
, xn
)
If E ∈ W 1,∞, then, the ﬁrst-order scheme possesses a unique solution such that
xn
ε − xε(tn)
E
(1 + t2
n ) e2Kx tn (1 + v0
ε + ε) min
∆t
ε3
(1 + ε2
), ∆t + ε .
If E ∈ W 2,∞:
yn
ε − yε(tn)
E
(1 + t4
n ) e2Kx tn (1 + v0
ε
2
+ ε2
) min
∆t
ε3
(1 + ε) , ∆t + ε2
,
if, for the ﬁrst step, we have either
∆t = O(ε)
fully-implicit method
11/19

Corollary
xn
ε − xε(tn)
(t,E, v0
ε )
(1 + ε)



∆t
ε3 (1 + ε2), if ∆t ≤ ε4 ,
ε, if ε4 ≤ ∆t ≤ ε ,
∆t, if ε ≤ ∆t ,
yn
ε − yε(tn)
(t,E, v0
ε )
(1 + ε2
)



∆t
ε3 (1 + ε), if ∆t ≤ ε5 ,
ε2, if ε5 ≤ ∆t ≤ ε2 ,
∆t, if ε2 ≤ ∆t .
12/19

Corollary
xn
ε − xε(tn)
(t,E, v0
ε )
(1 + ε)



∆t
ε3 (1 + ε2), if ∆t ≤ ε4 ,
ε, if ε4 ≤ ∆t ≤ ε ,
∆t, if ε ≤ ∆t ,
yn
ε − yε(tn)
(t,E, v0
ε )
(1 + ε2
)



∆t
ε3 (1 + ε), if ∆t ≤ ε5 ,
ε2, if ε5 ≤ ∆t ≤ ε2 ,
∆t, if ε2 ≤ ∆t .
Worst-case scenario
xn
ε − xε(tn)
(t,E, v0
ε )
(∆t)1/4
, yn
ε − yε(tn)
(t,E, v0
ε )
(∆t)2/5
,
12/19

Proof
Similar to continuous case
Taylor expansion:
yn+1
ε − yn
ε
∆t
= − E⊥
(tn, yn
ε ) − ε dx E⊥
(tn, yn
ε )(vn
ε )⊥
+ ε2
Θ0(tn, yn
ε , vn
ε ) , n ≥ 0 ,
13/19

Proof
Taylor expansion:
yn+1
ε − yn
ε
∆t
= − E⊥
(tn, yn
ε ) − ε dx E⊥
(tn, yn
ε )(vn
ε )⊥
+ ε2
Θ0(tn, yn
ε , vn
ε ) , n ≥ 0 ,
employ the velocity update to get
dx E(tn, yn
ε )(vn
ε )⊥
= −ε2
dx E(tn, yn
ε )
vn
ε − vn−1
ε
∆t
−
1
ε
E(tn−1, xn−1
ε )
13/19

Proof
Taylor expansion:
yn+1
ε − yn
ε
∆t
= − E⊥
(tn, yn
ε ) − ε dx E⊥
(tn, yn
ε )(vn
ε )⊥
+ ε2
Θ0(tn, yn
ε , vn
ε ) , n ≥ 0 ,
dx E(tn, yn
ε )(vn
ε )⊥
= −ε2
dx E(tn, yn
ε )
vn
ε − vn−1
ε
∆t
−
1
ε
E(tn−1, xn−1
ε )
classical analysis:
Lemma (consistency)
τn
y
E
∆t
ε
eKx tn+1 v0
ε + ε (1 + tn+1)
τn
v
E
∆t
ε4
eKx tn+1 (1 + ε2
) v0
ε + ε (1 + tn+1)
13/19

Proof
Taylor expansion:
yn+1
ε − yn
ε
∆t
= − E⊥
(tn, yn
ε ) − ε dx E⊥
(tn, yn
ε )(vn
ε )⊥
+ ε2
Θ0(tn, yn
ε , vn
ε ) , n ≥ 0 ,
dx E(tn, yn
ε )(vn
ε )⊥
= −ε2
dx E(tn, yn
ε )
vn
ε − vn−1
ε
∆t
−
1
ε
E(tn−1, xn−1
ε )
classical analysis:
Lemma (consistency)
τn
y
E
∆t
ε
eKx tn+1 v0
ε + ε (1 + tn+1)
τn
v
E
∆t
ε4
eKx tn+1 (1 + ε2
) v0
ε + ε (1 + tn+1)
yε(tn) − yn
ε + ε vε(tn) − vn
ε ≤
n−1
k=0
e2 Kx (tn−tk+1)
τk
y + ε τk
v ) ∆t
13/19

Modiﬁed ﬁrst-order IMEX scheme



yn+1
ε − yn
ε
∆t
= −E⊥(tn, yn
ε + ε(vn+1
ε )⊥)
ε
vn+1
ε − vn
ε
∆t
= E(tn, yn
ε + ε(vn
ε )⊥) −
1
ε
(vn+1
ε )⊥
14/19

Modified first-order IMEX scheme



yn+1
ε − yn
ε
∆t
= −E⊥(tn, yn
ε + ε(vn+1
ε )⊥)
ε
vn+1
ε − vn
ε
∆t
= E(tn, yn
ε + ε(vn
ε )⊥) −
1
ε
(vn+1
ε )⊥
If E ∈ W 1,∞, then, the first-order scheme possesses a unique solution such that
xn
ε − xε(tn)
E
(1 + t3
n ) e(2+Kx ∆t)Kx tn (1 + v0
ε + ε) min
∆t
ε3
(1 + ε2
), ∆t + ε
If E ∈ W 2,∞:
yn
ε − yε(tn)
E
(1 + t4
n ) e(2+Kx ∆t)Kx tn (1 + v0
ε
2
+ ε2
) min
∆t
ε3
(1 + ε) , ∆t + ε2
.
14/19

Outline
Introduction
Discrete estimates
Numerical example
14/19

Numerical example
electric potential: φ(x) = 1
2
x 2 + 1
10π
cos2(2πx2)
uniform magnetic ﬁeld: B(t, x) = 1
ε
(0, 0, 1)T
“ill-prepared” initial condition
x0
ε = (1, 1)
v0
ε = (3, 3)
15/19

Numerical example
electric potential: φ(x) = 1
2
x 2 + 1
10π
cos2(2πx2)
uniform magnetic ﬁeld: B(t, x) = 1
ε
(0, 0, 1)T
“ill-prepared” initial condition
x0
ε = (1, 1)
v0
ε = (3, 3)
Measure the error:



Ey (∆t, ε) :=
NT
n=1
∆t yn
ε − yε(tn)
Ey, gc(∆t, ε) :=
NT
n=1
∆t yn
ε − X(tn, 0, x0
− ε(v0
ε )⊥
)
15/19

Modiﬁed ﬁrst-order method
10-4
10-3
10-2
10-1
100
101
10-5 10-4 10-3 10-2 10-1 100
slope of 2
εy
ε in log scale
Δt = 0.0001
Δt = 0.0004
Δt = 0.0016
Δt = 0.0064
Δt = 0.0256
Δt = 0.1024
10-4
10-3
10-2
10-1
100
101
10-5 10-4 10-3 10-2 10-1 100
slope of 2
εy,gc
ε in log scale
Δt = 0.0001
Δt = 0.0004
Δt = 0.0016
Δt = 0.0064
Δt = 0.0256
Δt = 0.1024
16/19

Second-order method
RK (for the explicit part) and an L-stable second-order SDIRK method (for the
implicit part) [Filbet and Rodrigues, 2016]
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
10-5 10-4 10-3 10-2 10-1 100
slope of 2
εy
ε in log scale
Δt = 0.0001
Δt = 0.0004
Δt = 0.0016
Δt = 0.0064
Δt = 0.0256
Δt = 0.1024
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
10-5 10-4 10-3 10-2 10-1
slope of 2
εy,gc
ε in log scale
Δt = 0.0001
Δt = 0.0004
Δt = 0.0016
Δt = 0.0064
Δt = 0.0256
Δt = 0.1024
17/19

Conclusion
We have analyzed IMEX-PIC scheme for Vlasov–Poisson equations (with a given E):
ε-uniform stability
uniform convergence estimates (continuous, discrete)
Perspectives
second-order schemes → O(ε3)-estimates
inhomogeneous magnetic ﬁeld [Filbet and Rodrigues, 2017]
velocity converges weakly to zero
kinetic energy converges strongly to non-zero!
3d [Degond and Filbet, 2016; Filbet et al., 2017]
Thanks For Your Attention!
18/19

References I
Albert Cohen and Benoit Perthame. Optimal approximations of transport equations by particle and
pseudoparticle methods. SIAM Journal on Mathematical Analysis, 32(3):616–636, 2000.
Pierre Degond and Francis Filbet. On the asymptotic limit of the three dimensional Vlasov–Poisson
system for large magnetic field: formal derivation. arXiv preprint arXiv:1603.03666, 2016.
Francis Filbet and Luis M. Rodrigues. Asymptotically stable particle-in-cell methods for the
Vlasov–Poisson system with a strong external magnetic field. SIAM Journal on Numerical Analysis,
54(2):1120–1146, 2016.
Francis Filbet and Luis Miguel Rodrigues. Asymptotically preserving particle-in-cell methods for
inhomogeneous strongly magnetized plasmas. SIAM Journal on Numerical Analysis, 55(5):
2416–2443, 2017.
Francis Filbet, Tao Xiong, and Eric Sonnendrücker. On the Vlasov–Maxwell system with a strong
external magnetic field. 2017.
Francis Filbet, L. Miguel Rodrigues, and Hamed Zakerzadeh. Convergence analysis of asymptotic
preserving schemes for strongly magnetized plasmas. In preparation, 2019.
Arlen M. Il’in. Differencing scheme for a differential equation with a small parameter affecting the
highest derivative. Mathematical Notes of the Academy of Sciences of the USSR, 6(2):596–602, 1969.
Shi Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM
Journal on Scientific Computing, 21(2):441–454, 1999.
Shi Jin. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review.
Lecture Notes for Summer School on “Methods and Models of Kinetic Theory” (M&MKT), Porto
Ercole (Grosseto, Italy), pages 177–216, 2010.
Edward W. Larsen, J. E. Morel, and Warren F. Miller. Asymptotic solutions of numerical transport
problems in optically thick, diffusive regimes. Journal of Computational Physics, 69(2):283–324, 1987.
19/19

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