dsmc methods for multicomponent plasmas
TRANSCRIPT
”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