”Methods & Models of Kinetic Theory”

Porto Ercole, June 3th - June 9th, 2012.

DSMC methods for multicomponent plasmas

A. V. Bobylev

Karlstad, Sweden

in collaboration with

I. F. Potapenko, S. A. Karpov

Moscow, Russia

Introduction

Kinetic equations for plasmas

fi(x, v, t), i = 1, .., n- distribution functions;

x ∈ R3 - position, v ∈ R3 -velocity, t ∈ R+ -time

Evolution equations: Difi =n∑

j=1QL

ij(fi, fj), where

Di = ∂t + v · ∂x +ei

mi

(E +

1

cv ×B

)· ∂v, i = 1, .., n

E(x, t) and B(x, t) are electric and magnetic vector fields

(external + self-consistent fields)

Standard splitting on [t0, t0 +∆t]:

(1) Difi = 0 ←− Vlasov equations

(2) ∂tfi =n∑

j=1QL

ij(fi, fj) ←− Landau (LFP) equations

Particle Methods:

(1) PiC for Vlasov

(2) DSMC for Landau

Landau Equation (LE)

Landau (1936) generalized Boltzmann equation to the case of Coulomb interaction

Uij =eiej

|xi − xj|- interparticle potential

Rough idea:

Consider a modified potential Uij = Uij exp(−rij/rD) , with the Debye radius rD, and

find the leading asymptotic term of the Boltzmann collision integral, as rD →∞.

The result reads

∂fi(v, t)

∂t= 2πL

n∑j=1

e2i e2j

m2i

∂

∂vα

∫R3

dwRαβ(v −w)

(∂

∂vβ−

mi

mj

∂

∂wβ

)fi(v)fj(w),

where

Rαβ(u) =u2δαβ − uαuβ

u3, α, β = 1, 2, 3, i, j = 1, ..., n,

L = log(rD/r0) - Coulomb logarithm

LE in Fokker-Planck form

Rosenbluth, MacDonald, and Judd (1957) re-discovered LE by postulating it in FP form.

The results reads

1

4πL

∂fi

∂t=

∂

∂vα

−fi

∂hi

∂vα+

1

2

∂

∂vβ

(fi

∂2gi

∂vα∂vβ

),

hi and gi are called Rosenbluth potentials

hi =n∑

j=1

Kij

∫R3

dwfj(w, t)| v −w |−1, gi =n∑

j=1

Kijmj

mi

∫R3

dwfj(w, t)| v −w |,

Kij =e2i e

2j

mimj

, i = 1, .., n

This form of LE is very useful for regular (deterministic) numerical methods.

Numerical methods for LE

(A) Deterministic methods

Long history since 1957, see I.Potapenko, A.B. and E.Mossberg, TTSP 37 (2008)

for a review (it contains more than 50 refs.)

(B) Stochastic (particle) methods

(B1) Simulation by Langevin-type stochastic DEs (nonlinear diffusion)

Disadvantage: roughly ∼ N2 operations, N is a number of

particles

(B2) DSMC methods, simulation of pair collisions, Bird-type methods

(linear in N)

This is the main motivation for (B2)

Monte Carlo methods for LE Key references:

• 1 Takizuka and Abe (1977);

• 2 Nanbu (1997);

• 3 B. and Nanbu (2000).

The methods of TA and N are based on heuristic (physical) arguments. In

particular, Nanbu (1997) does not use any kinetic equation.

Main result of 3 :

systematic derivation of MC methods from kinetic

equations

and a lot of new numerical schemes.

The simplest new scheme: B., Mossberg and Potapenko (2006)

More recent contributions: series of papers by Caflish, Pareschi, and

co-authors (2008-2011). In particular, Dimarco, Caflish and Pareschi (2010) (an

important step to understanding)

The simplest DSMC scheme for one LE

(B., Nanbu - 2000, B., Mossberg, Potapenko - 2006)

At time t we have N velocities: VN(t) = v1(t), ..., vN(t) ∈ R3N .

How to find VN(t+∆t)? (of course N →∞, ∆t→ 0).

Algorithm

1. Choose any pair (vi, vj), i < j and perform a collision, i.e. set

v′i =1

2(vi + vj + |u|ω), v′j =

1

2(vi + vj − |u|ω), u = vi − vj

where ω = (θ, φ) ∈ S2 is a unit vector (in spherical coordinates with z-axis along u,

having randomly distributed φ ∈ [0, π] and the scattering angle θ ∈ [0, π] given by

cos θ = 1−Min

(a∆t

|u|3, 2

), a = const )

2. Set vi = v′i, vj = v′j and repeat step 1.

3. Then, after N collisions, you obtain VN(t+∆t) .

Comments on the scheme

1. It is extremely simple because

(a) pairs (vi, vj) are taken randomly, like for Maxwell molecules (total collision

frequency does not depend on velocities)

(b) scattering angle is given explicitly (this is a main difference from schemes

by TA and by N. They use much more complicated ”scattering laws”).

2. Scattering angle depends on the time step ∆t. Roughly,

θ2 ≃2 a ∆t

|u|3, ∆t→ 0.

Question. Why does this scheme work for LE?

Hint. LE and BE - how are they connected?

Main step: Approximation of Landau equations by Boltzmann equations

Begin with a mixture of n ≥ 1 neutral gases with masses mi, i = 1, .., n:

∂fi

∂t=

n∑j=1

Qij(fi, fj), i = 1, ..., n,

where

Qij(fi, fj) =

∫R3×S2

dw dω gij(u,

u ·ωu

)[fi(v

′)fj(w′)− fi(v)fj(w)] , i, j = 1, ..., n,

u = v −w, ω ∈ S2, u = |u|, v′ =1

mi +mj

(miv +mjw +mjuω) ,

gij(u, µ) = gji(u, µ) = uσij(u, µ), w′ =1

mi +mj

(miv +mjw −miuω) ;

σij(u, µ) - differential cross-section, µ = cos θ, θ ∈ [0, π] - scattering angle

Remark.

For brevity we assume below that the functions fi(v) and gij are ”as good as we need”

Proposition 1 (B., 1975)

Boltzmann integral Qij(fi, fj) can be expanded in formal series

Qij(fi, fj) =

∞∑k=1

Q(k)ij (fi, fj),

where the first term corresponds to the Landau collision integral (for arbitrary gij(u, µ)):

Q(1)ij (fi, fj) =

m2ij

2mi

∂

∂vα

∫R3

dw g(1)ij (u)Tαβ(u)

(1

mi

∂

∂vβ−

1

mj

∂

∂wβ

)fi(v)fj(w),

mij =mimj

mi +mj, Tαβ(u) = u2δαβ − uαuβ, g

(1)ij (u) = 2π

1∫−1

dµ gij(u, µ)(1− µ).

The other terms can be symbolically written in the form

Q(k)ij (fi, fj) =

∫R3

dw g(k)ij (u)A

(k)ij (v,w), g

(k)ij (u) = 2π

1∫−1

dµ gij(u, µ)(1− µ)k, k ≥ 2,

where A(k)ij (v,w) is a smooth integrable function for any k ≥ 2

Generalized Landau equations ( for arbitrary ”cross-section” gij(u, µ) )

∂fi

∂t=

n∑j=1

Q(1)ij (fi, fj), i, j = 1, ..., n, where

Q(1)ij (fi, fj) =

m2ij

2mi

∂

∂vα

∫R3

dw g(1)ij (u)Tαβ(u)

(1

mi

∂

∂vβ−

1

mj

∂

∂wβ

)fi(v)fj(w).

Our problem:

given Landau equations with g(1)ij (u) = bij(u) = 4πL

e2i e2j

m2ij

u−3; i, j = 1, ..., n.

How to approximate them by BE?

Answer:

Take some functions gij(u, µ; ε) ≥ 0 such that

limε→0

2π

1∫−1

dµ gij(u, µ; ε)(1− µ) = bij(u), limε→0

2π

1∫−1

dµ gij(u, µ; ε)(1− µ)k = 0, k ≥ 2.

The simplest example of the approximation

g(k)ij (u, µ; ε) =

1

2πεδ 1− µ−Min [ε bij(u), 2] .

Then

g(k)ij (u; ε) = 2π

1∫−1

dµ gij(u, µ; ε)(1− µ)k =1

εMin [ε bij(u), 2]k −−→

ε→00 if k > 1.

Note that

g(0)ij = 1/ε, g

(1)ij = Min [bij(u), 2ε−1] −−→

ε→0bij(u) .

⇓

⇓ quasi-Maxwellian approximation

(total collision frequencies are independent of velocities)

Comment.

This property yields an important simplification of all DSMC methods:

⇓ colliding pairs can be chosen randomly.

General class of quasi-Maxwellian approximation

(for simplicity - one Landau equation)

∂f(v, t)

∂t= QL(f, f) =

∂

∂vi

∫R3

dwu2δij − uiuj

u3

(∂

∂vj−

∂

∂wj

)f(v)f(w); i, j = 1, 2, 3.

We approximate it by the Boltzmann equation

∂f

∂t= Qε(f, f) =

∫R3×S2

dw dω gε(u,

u ·ωu

)[fi(v

′)fj(w′)− fi(v)fj(w)

],

u = v −w, ω ∈ S2, v′ =1

2(v +w + uω) , w′ =

1

2(v +w − uω) .

Here gε(u, µ) =1

2πεΨ(s, 1− µ), s = 8ε

u3 ,

then g(k)(u; ε) = ε−1 Ψk(s), Ψk(s) =2∫0

dxΨ(s, x)xk,

In addition we assume that

(1) Ψ0(s) = 1− quasi-Maxwellian approximation

(2) lims→0

1

sΨ1(s) = 1− this is needed for LE

Conclusions

There are infinitely many ways to construct such an approximation.

They differ just by functions Ψ(s, x). In particular,

1. our (simplest) method (2006):

Ψ(s, x) = δ[x−Min(s, 2)], 0 < x ≤ 2

2. Nanbu method (1997):

ΨN(s, x) =A

1− e−2Ae−Ax, A = A(s), cthA−A−1 = e−s.

3. Takizuka & Abe method (1977):

ΨTA =[π3sx(2− x)3

]−1/2exp

[−

x

s(2− x)

].

Order of approximation

Problem: Boltzmann integral Qε(f, f) with kernel

gε(u, µ) =1

2πεΨ(s, 1− µ), s =

8ε

u3,

Ψk(s) =

2∫0

dxΨ(s, x)xk, k ≥ 0.

What can be said about the difference

∆ε(v) = Qε(f, f)−QL(f, f), ε→ 0,

for some reasonable functions f(v)?

Class of functions There exist two positive numbers A and β such that

|f (m)(v)| ≤ Ae−βv2

; m = 0, 1, 2, 3, v ∈ R3,

where f (m)(v) denotes all partial derivatives of order m.

Proposition 2 If these conditions are satisfied and

(a) |Ψ1(s)− s| = O(s2), |Ψ2(s)| = O(s2), s→ 0,

(b)

2∫0

dx Ψ(s, x) x3/2 exp[β (ε/s)2/3 x

]= O(s3/2), s→ 0,

then

|Qε(f, f)−QL(f, f)| ≤ C√ε e−βv

2

(1 + v).

Comment.

These conditions are easy to verify for any given Ψ(s, x).

In particular, they are fulfilled for functions ΨTA(s, x) and ΨN(s, x)

and for our function Ψ = δ[x−Min(s, 2)].

Similar statement holds for the case of mixture (different masses).

True order of approximation? Probably O(εα), 1/2 ≤ α ≤ 1.

DSMC method

(for brevity - one Landau equation)

We assume that we chose to approximate LE by BE with given gε(u, µ),

i.e.

gε(u, µ) =1

2πεδ

[1− µ−Min

(8ε

u3, 2

)],

BE reads

f(v, t) :∂f

∂t= Qε(f, f), f|t=0 = f0.

How to solve it?

Main references:

G.A. Bird (1968-70)

M. Kac (1957)

Stochastic Model (M. Kac, 1957)

VN = v1(t), ...,vN(t) ∈ R3N .

Time evolution of VN : by jumps (pair collisions)

VN → V′(i,j)

N = vi, ...v′

i, ...,v′

j, ...,vN

Then we introduce a distribution function φ(VN , t) and obtain Master equation

∂ φ(VN)

∂t=

1

N

N−1∑i=1

N∑j=i+1

L(i|j) φ(VN),

where

L(i|j) φ(VN) =

∫S2

dω gε(u,

u ·ωu

)φ[V

′(i,j)

N ]− φ(VN),

u = vi − vj, V′(i,j)

N = ...v′i, ...,v′j, ...

Rough connection with BE:

f(v1) =

∫R3(N−1)

dv2...vN φ(VN), F (v1,v2) =

∫R3(N−2)

dv3...vN φ(VN).

Then∂f

∂t=

N − 1

N

∫R3×S2

dwdΩ gε(u,

u ·ωu

)[F (v′,w′) − F (v,w)] (*)

It remains to set

F (v,w) ≃ f(v) f(w) (molecular chaos)

in the limit N →∞.

Then (*) becomes BE.

How to solve Master Equation?

ME can be written as

ME can be written as φt =1

NLN φ, LN = L+

N − L−N ,

where

L−N =N (N − 1)

2gtot(u), gtot(u) =

∫S2

dω gε(u,

u ·ωu

)=

1

ε,

i.e. L−N = νN = const. for quasi-Maxwellian model. Hence, we obtain

φt =1

NL+N φ− νN φ, νN =

N (N − 1)

2 ε.

Then the approximation

φt ≈φ(t+∆t) − φ(t)

∆t

leads to equality

φ(t+∆t) =∆t

NL+N +

(1 −

νN ∆t

N

)φ .

It remains to choose

∆t = ∆tmax =N

νN=

2 ε

N − 1≈

2 ε

N.

Finally we obtain

φ(VN , t+ τN) =1

νNL+N φ(VN , t) = ⟨φ(V′N , t)⟩ τN =

N

νN=

2 ε

N − 1.

The averaging ⟨...⟩ is made over all possible pairs and all possible results of each collision.

Thus, in our model the scattering angle θ is given by

cos θ = µ = 1 − Min

(8ε

u3, 2

)

and the interval between two collisions is

τ ≃2ε

N.

If we denote ∆tmacro = 2ε, then we obtain the above described DSMC algorithm

with a = 4.

Initial isotropic and anisotropic functions in Cartesian coordinates

-1,0-0,5

0,00,5

1,0

-1,0

-0,5

0,0

0,5

1,0

-1,0

-0,50,0

0,51,0

VZ

V YVX

-1,0-0,5

0,00,5

1,0

-1,0

-0,5

0,0

0,5

1,0

-1,0

-0,50,0

0,51,0

VZ

V YVX

Evolution of moments on time:

1 - the 2-nd ion moment, 2 - the 4-th ion moment, and the 2-nd electron moment.

λ = 1, κ = 1 and λ = γ1/2, κ = 1/4, γ = 1/64, N1 = 500, K = 20, ε = 0.05.

0,0 0,5 1,0 1,50,4

0,6

0,8

1,0

ion

dist

ribut

ion

func

tion

mom

ents

time

difference scheme = 1, = 1 = , = 1/4

x104

1

2

0,0 0,5 1,0 1,50

5

10

15

20

25

30

35

difference scheme = 1, = 1 = , = 1/4

elec

tron

dist

ribut

ion

func

tion

mom

ents

time x104

Evolution of moments in isotropic case:

1 - the 2-nd ion moment, 2 - the 4-th ion moment, and the 2-nd electron moment.

λ = 1, κ = 1 and λ = γ1/2, κ = 1/4 at γ = 1/1800, N1 = 1000, K = 20, ε = 0.05.

0,0 0,2 0,4 0,6 0,8 1,00,92

0,94

0,96

0,98

1,00 = 1, = 1 = , = 1/4

ion

dist

ribut

ion

func

tion

mom

ents

time x104

1

2

0,0 0,2 0,4 0,6 0,8 1,0

0,0

0,2

0,4

0,6

0,8

1,0

elec

tron

dist

ribut

ion

func

tion

mom

ents

time

= 1, = 1 = , = 1/4

x104

x104

Relaxation of the anisotropic initial functions -

fe, fi ∼ δ(v − 1)δ(µ),meT 0

i

miT 0e= 1, T 0

z = 0

Electron temperature Electron and ion temperatures

0 10 20 30

0

1

2

3

elec

tron

tem

pera

ture

time

1

2

0,0 0,5 1,0 1,5

0

5

10

15

20

25

30

35

tempe

ratures

time x104

4

32

1

”Runaway” electrons (and not only). Long-range potentials: U ∼ 1/rβ, 1 ≤ β < 4

β = 1 - Coulomb interaction, β = 2 - dipole interaction, β = 4 - Maxwellian molecules.

∂fe/∂t+ Ez ∂fe/∂vz = Q(fe, fe) +Q(fe, fi), Ez = 0.1, vz(t+∆t)→ vz(t) + Ez∆t

β = 1, dashed line - one sort of particles, Two sorts of particles, β = 4, 3, 2, 1

solid line - electron and ions.

0 10 20 30 40 500

2

4

6

8

10

elec

tron

dist

ribut

ion

func

tion

mom

ents

time

1

2

3

4

0 20 40 60 80 100

0,0

0,1

0,2

0,3

0,4

0,5

elec

tron

curr

ent

time

1

2

3

4

”Runaway” electron distribution functions with < v >= 0.8

one sort of particles electron and ions

-4-2

02

4

-1

0

1

2

3

4

5

-4-2

02

4

VZ

V YVX

-4-2

02

4

-1

0

1

2

3

4

5

-4-2

02

4

VZ

V YVX