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.