Particle-In-Cell approximation to the Vlasov-Poisson with astrong external magnetic field
Francis FILBET
Institut de Mathématiques de Toulouse
Université Toulouse III & Institut Universitaire de France
Mathematical Topics in Kinetic TheoryCambridge University, May 2016
Example of a single particle motion
We consider only one charged particle submitted to an intenseelectromagnetic field
εdxdt
= v,
εdvdt
= E(t , x) +1ε
v× B(t , x),
where E and B are given and non uniform.
Space and velocity for ε = 0.2
Outline of the Talk
Research programUnderstand how to recover the correct guiding center velocity at the levelof the kinetic equation
Preserve the structure of the Vlasov-Poisson system in phase space andconstruct numerical scheme on the original problem not on gyrokineticmodels.
1 Part I. Modeling and scaling issuesVlasov-Poisson system with a strong external magnetic field2D problem3D problem
2 Part II. Strategy for numerical schemesDefinition and state of the artA class of semi-implicit schemes for particle methods
3 Numerical simulationsNumerical simulations: large magnetic fieldsVlasov-Poisson with a large magnetic field
Vlasov-Poisson system with a strong external magnetic field
AssumptionsConsider the Vlasov-Poisson system 3D × 3D
Consider that the magnetic field is uniform Bext = bext(t , x) ez , where ez
stands for the unit vector in the z-direction,
we are interesting by the long time asymptotic of electrons
After a rescaling, it yieldsε∂f∂t
+ v · ∇xf +
[E +
bext(t , x)
εv⊥]· ∇vf = 0.
E = −∇φ, −∆φ = ρ− ρi .
whereρ =
∫R3
fdv.
From the works of Arsenev, or DiPerna-Lions, there exist global in time weaksolutions (energy and Lp estimates are uniform with respect to time and ε).→ this framework allows us to study the asymptotic ε→ 0.
In the limit ε→ 0 for the 2D problem
First case : bext = 1.If we are not interesting by the details of the dynamics of electrons, we onlywant the evolution of the density ρ.
We defineρε :=
∫R2
f ε dv, and Jε :=
∫R2
f εv dv,
and get ∂ρε
∂t+
1ε
divx(Jε) = 0
ε∂Jε
∂t+ divx
∫Rd
v× v f ε dv + E ρ +1ε
Jε,⊥ = 0
Then1 the leading term induces that f (v) ' F (‖v‖),2 in the limit ε→ 0
1ε
Jε →(∇x
∫Rd‖v‖2 F dv + E ρ
)⊥
In the limit ε→ 0 for the 2D problem
It gives the guiding center system U = E⊥ = −∇⊥φ∂ρ
∂t+ divxUρ = 0
−∆φ = ρ− ρi .
This model satisfies some basic properties1 preservation of energy
ddt
∫‖E(t , x)‖2 dx = 0.
2 incompressible flow divxU = 0.3 preservation of Lp norm of the density and for any continuous function η
ddt
∫η( ρ(t , x) ) dx = 0.
This asymptotic limit is justified rigorously by F. Golse and L. Saint-Raymondfor weak solutions 1.
1Golse-St-Raymond, JMPA’0, St-Raymond, JMPA’02
In the limit ε→ 0 for the 2D problem
Second case : bext ≡ b(t ,x).We cannot get an equation on the density ρ but on the distribution functionF ≡ F (t , x,w) with w = ‖v‖.
1 From a Hilbert expansion of f ε = f0 + ε f1 + ...
2 Apply a change of coordinate v = w ew (θ).3 We proceed as before but now only integrate on the angular velocityθ ∈ [0, 2π].
We formally get f0 = F such that
∂F∂t
+ U · ∇xF + uw∂F∂w
= 0,
where U corresponds to the drift velocity and (U, uw ) is given by
U =1b
(E − w2
2b∇x⊥b
)⊥, uw =
w2b2 ∇
⊥x b · E,
The drift-velocity results from two drifts,Energy structure is preserved and the flow remains incompressible.
Idea of the proof and open problem
Consider that (E,B) are given and smooth. We define
F ε(w) =1
2π
∫ 2π
0f εdθ, Jε(w) =
12π
∫ 2π
0ew (θ) f ε dθ.
and
Σε =1
2π
∫ 2π
0
(ew (θ)⊗ ew (θ)− 1
2Id)
f ε dθ
Then, we get the following equation
∂tF ε − divx
(w2
2∇⊥F ε +
w2
E⊥ ∂w F ε)
− 1w∂w
(w2
2∇⊥x F ε · E +
w2
E⊥ · E∂w F ε)
= Rε,
where Rε is given by
Rε = w divx (εw ∂tJε + wdivx (Σε) + ∂w Σε E + 2 Σε E)⊥
+1w∂w
(w (εw ∂tJε + wdivx (Σε) + ∂w Σε E + 2Σε E )⊥ · E
).
Open problem : Vlasov-Poisson system.
In the limit ε→ 0 for the 3D problem
Considering the 3D Vlasov-Poisson system with an external magnetic field, itis possible to get an asymptotic model for (F ,P).Applying the same strategy, we get2 (for a uniform magnetic field)
∂F∂t
+ E⊥⊥ · ∇x⊥F + v‖∂P∂x‖
+ E‖∂P∂v‖
= 0,
v‖∂F∂x‖
+ E‖∂F∂v‖
= 0,
where E = (E⊥,E‖) is the electric field.
PropertiesThis model preserve the fundamental properties of the Vlasov-Poissonsystem : energy conservation, divergence free flow, positivity of F .
Furthermore, we recover the classical drift velocity (E×B, ∇|B| ×B, etc)
Adiabatic invariance.
2FF and P. Degond, arXiv:0905.2400 (2016)
Approximation of multi-scale problems
We first introduce the consept of Asymptotic Preserving schemes. It is basedon the work of S. Jin and A. Klar3
1 Uniform stability. We want to use a large time step h = ∆t even whenε 1. For that we are ready to lose microscopic informations comingfrom f since in our case ρ is the appropriate quantity.
2 Uniform consistency. We want to preserve the order of accuracy of thelimit ε→ 0. Observe that in general splitting schemes do not satisfy sucha property.
3S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations(1999).
Particle methods
The particles method consists in approximating the initial condition f0 by thefollowing Dirac mass sum
f 0N(x, v) :=
∑1≤k≤N
ωk δ(x− x0k ) δ(v− v0
k ) ,
where (x0k , v
0k )1≤k≤N is a beam of N particles distributed in the four
dimensional phase space according to the density function f0. Afterwards,one approximates the solution of Vlasov, by
fN(t , x, v) :=∑
1≤k≤N
ωk δ(x− Xk (t)) δ(v− Vk (t)) ,
where (Xk ,Vk )1≤k≤N is the position in phase space of particle k movingalong the characteristic curves with the initial data (x0
k , v0k ), for 1 ≤ k ≤ N.
For the computation of the electric field, the Dirac mass has to be replaced bya smooth function ϕα
fN,α(t , x, v) :=∑
1≤k≤N
ωk ϕα(x− xk (t)) ϕα(v− vk (t)) ,
where ϕα = α−dϕ(·/α) is a particle shape function
Particle methods
The classical error estimate reads then as follows 4:
Proposition
Consider the Vlasov equation with a given electromagnetic field (E,Bext) anda smooth initial datum f 0 ∈ Cs
c (Rd ), with s ≥ 1.If for some prescribed integers m > 0 and r > 0, the cut-off ϕ ≥ 0 has m-thorder smoothness and satisfies a moment condition of order r , namely,∫
Rdϕ(y) dy = 1,
∫Rd|y|r ϕ(y) dy < ∞,
and∫Rd
y s11 . . . y sd
d ϕ(y) dy = 0, for s ∈ Nd with 1 ≤ s1 + · · ·+ sd ≤ r − 1.
Then there exists a constant C independent of f0, N or α, such that we havefor all 1 ≤ p ≤ +∞,
‖f (t)− fN,α(t)‖Lp → 0,
when N →∞ and α→ 0 where the ratio N1/dα 1.
4Cohen & Perthame 2000
Brief bibliography
Aim. Construct a numerical scheme wich preserves the asymptotic behaviorwhen ε→ 0.
DifficultyThere is no relaxation limit and no relaxation process to a unique equilibrium.
Concerning the Vlasov-Poisson system with an external magnetic field
N. Crouseilles, M. Lemou and F. Méhats, Asymptotic preserving schemes forhighly oscillatory Vlasov-Poisson equation, JCP (2013).
E. Frénod, S.A. Hirstoaga, M. Lutz, E. Sonnendrücker, Long timebehaviour of an exponential integrator for a Vlasov-Poisson system with strongmagnetic field, CiCP (2015)
Other related works
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical IntegrationStructure-Preserving Algorithms for Ordinary Differential Equations
Ph. Chartier, E. Faou, group IPSO in Rennes
Particle-In-Cell approximation
Let us now consider the Vlasov-Poisson system with an external magneticfield and the corresponding characteristic curves which read
εdXdt
= V,
εdVdt
=bext(t ,X)
εV⊥ + E(t ,X),
X(t0) = x0, V(t0) = v0,
(1)
where the electric field is computed from the Poisson equation.
The Particle-In-Cell method,
we consider a set of particles characterized by a weight (wk)k∈N andtheir position in phase space (xn
k, vnk)k∈N computed by discretizing the
Vlasov-Poisson system at time tn = n ∆t .
the solution f is discretized as follows
f n+1h (x, v) :=
∑k∈ZZd
wk ϕh(x− xnk)ϕh(v− vn
k),
A class of semi-implicit schemes
We establish a large class of high order semi-implicit schemesa fordudt
(t) = H(t , u(t), u(t)), ∀ t ≥ t0,
u(t0) = u0,
(2)
with H: R× Rm × Rm → Rm sufficiently regular
Dependence on the second argument of H is non stiff.
Dependence on the third argument is stiff.aBoscarino, Filbet and Russo, J. Sci. Comput. (2016)
Consider the first order Euler semi-implicit scheme
εXn+1 − Xn
∆t= Vn+1,
εVn+1 − Vn
∆t=
bext(tn,Xn)
εVn+1,⊥ + E(tn,Xn),
X0 = x0, V0 = v0 .
(3)
Uniform accuracy result for large time step and small ε > 0
We want to compare our discrete solution to
yn+1 − yn
∆t=
E⊥(tn, yn)
bext(tn, yn). (4)
Theorem
Consider that the electric field is given E ∈ W 1,∞((0,T )× T2) and setλ := ∆t/ε2 and R[W] = W⊥/bext. Then,
Zn = ε−1Vn − R [E(tn−1,X n−1)]
satisfies
Zn+1 = [Id− λR]−1(Zn − R[E(tn,X n)− E(tn−1,X n−1)]) , n ≥ 1.
Moreover, there exists C > 0 such that
‖Xn − Yn‖ ≤ C ε2[1 +
∥∥∥∥1ε
V0 −R[E(t0,X 0)]
∥∥∥∥] eKx n∆t .
Towards plasma physics : one single particle motion ε = 0.1
Towards plasma physics : one single particle motion ε = 0.01
Towards plasma physics : one single particle motion ε = 0.1
Towards plasma physics : one single particle motion ε = 0.05
Towards plasma physics : one single particle motion ε = 0.01
Towards plasma physics : one single particle motion ε = 0.001
The Vlasov-Poisson system
We consider bext = 1 and solve the following systemε∂f∂t
+ v · ∇xf −[E +
bext(x)
εv⊥]· ∇vf = 0.
E = −∇φ, −∆φ = ρ0 − ρ.
whereρ =
∫R2
f dv.
We choose
the domain Ω as a disk D(0, 12)
a perturbation of an equilibrium as initial datum :
f0(x, v) = 1A(x) exp(−|v|2/2)/2π; x ∈ Ω, v ∈ R2
The Vlasov-Poisson system ε = 0.01
The Vlasov-Poisson system ε = 0.8
The Vlasov-Poisson system
We consider bext = 1 and solve the following systemε∂f∂t
+ v · ∇xf −[E +
bext(x)
εv⊥]· ∇vf = 0.
E = −∇φ, −∆φ = ρ0 − ρ.
whereρ =
∫R2
f dv.
We choose
the domain Ω is a square (0, 10)2
the initial distribution is
f0(x, v) = (exp(−r 21 /2)+exp(−r 2
2 /2)) exp(−|v|2/2)/8π2; x ∈ Ω, v ∈ R2
with r1 = |x− x1| and r2 = |x− x2|, with x1 = (3.5, 5) and x2 = (6.5, 5).
The Vlasov-Poisson system ε = 0.01
Conclusion
Comments :
Dominant term is a magnetics field 1ε
(v × B) · ∇v f , no more dissipativeeffects
We have performed a rigorous analysis on the particle trajectories, butstill the interpretation at the level of the kinetic model has to beunderstood.
Current and future works :Applications in plasma physics
Treat more complex problems : capture drift due to the gradients of themagnetic field, etc
Applications to numerical analysisBetter understanding of the stability of high order schemes for PIC methodsand for the Vlasov-Poisson system.
