LATEX TikZposter
Particle in Fourier Discretisation of Kinetic Equations
Jakob Ameres, Eric Sonnendrucker
Numerical Methods for Plasma Physics, Technische Universitat Munchen, Max-Planck-Institut fur Plasmaphysik
Particle in Fourier Discretisation of Kinetic Equations
Jakob Ameres, Eric Sonnendrucker
Numerical Methods for Plasma Physics, Technische Universitat Munchen, Max-Planck-Institut fur Plasmaphysik
Introduction
• The gyrokinetic model, which approximates the Vlasov-Maxwell equa-tions by averaging over the gyro-motion, is well suited for the study ofturbulent transport in tokamaks and stellarators. Gyrokinetic Particlein Cell (PIC) codes using a finite element (FEM) field description areknown to conserve energy but not momentum.
• Using the Vlasov-Poisson equations in periodic domains with a purelyFourier based field solver yields a Monte Carlo particle method, Particlein Fourier (PIF), conserving both energy and momentum.
• Fourier filters on FEM/PIC solvers are applied since the total number ofphysically relevant Fourier modes remains small. In the case of PIF onedirectly calculates the relevant modes without computational overhead.
• In the scope of a field aligned description we derive a field solver, whichcouples a two dimensional Fourier transform in the torus’ angular direc-tions to B-splines over the radial coordinate yielding a hybrid PIC/PIFscheme.
Particle Discretisation ofVlasov-Poisson
• The Vlasov equation with external magnetic Field B, div(B) = 0,
∂f
∂t+ v · ∇xf − (E + v ×B) · ∇vf = 0 (1)
with the Poisson equation for the Electric potential Φ
−∆Φ = ρ− 1, ρ =
∫f dv E := −∇Φ (2)
• Solve by following the characteristics (V (t), X(t)).
d
dtV (t) = − (E(t,X(t)) + V (t)×B(t,X(t))) ,
d
dtX(t) = V (t)
• The solution f of equation (1) is constant along any characteristic
f (t = 0, X(t = 0), V (t = 0)) = f (t,X(t), V (t)) ∀t ≥ 0
• Probability density g(t, x, v) with∫∫
g(t = 0, x, v) dx dv = 1,g(t = 0, x, v) ≥ 0 ∀(x, v)
∂g
∂t+ v · ∇xg − (E + v ×B) · ∇vg = 0.
With suppg ⊂ suppf, g is used to sample from f
• Let one characteristic (X(0), V (0)) be randomly distributed accordingto g(t = 0, ·, ·) and let the (xk, vk) be independent and identicallydistributed according to g(t = 0, ·, ·) for all k = 1, . . . , Np.
• Define constant weights and the particle discretisation fh of f .
ck :=f (t, xk(t), vk(t))
g(t, xk(t), vk(t))=f (t = 0, xk(0), vk(0))
g(t = 0, xk(0), vk(0))
f (t, x, v) ≈ fh(t, x, v) =1
Np
Np∑k=1
δ (x− xk(t)) δ (v − vk(t)) ck
Poisson Equation in Fourier Space
• The n-th spatial Fourier mode of ρ =∫f dv is
ρn(t) :=
∫ L
0einx
2πL ρ(x, t) dx =
∫R
∫ L
0einx
2πL ρ(x, t) dxdv
• Estimate ρn(t) from the particle discretisation fh(x, t)
ρk(t) ≈ ˆρk(t) :=
∫ ∫eikxfh(x, v, t) dxdv
=1
Np
Np∑k=1
ctk
∫ ∫eikxδ(x− xtn)δ(v − vtn) dxdv
=1
Np
Np∑k=1
ctkeikxtn
• The Electric Field is determined in Fourier space by the estimatedFourier modes.
E(x, t) =∑n6=0
e−inx2πL En(t) with En(t) =
ρn(t)
in≈
ˆρn(t)
in
• Momentum conservation at discrete level∫E(t, x)ρ(x, t) dx ≈ 1
Np
Np∑l=1
Nh∑n=−Nh,n6=0
1
Np
Np∑k=1
einxtkiclne−inx
tl
=1
N2p
Nh∑n=1
i
n
Np∑k,l=1
ein(xtk−xtl) − e−in(xtk−x
tl)
=−2
N2p
Nh∑n=1
Np∑k,l=1
1
nsin(n(xtk − x
tl))
︸ ︷︷ ︸−sin(n(xtl−x
tk))
= 0
Variance Reduction
• To reduce the variance of the estimate for the fields the δf method[1] is used.∫
f1 dx∫∫
eixf2 dx∫∫
eixf3 dxdv∫∫
eixf4 dxdv
Estimating integrals with Np = 100 randomly distributed markers,uniformly in x and normally in v and the standard Monte Carloestimator θ. Introduction of a control variate h allows sampling thedifference δf = f − h thus reducing variance.
• Step function f1(x) :=⌊x
8
⌋, h1(x) = x
• For a small perturbation f2(x) := 1+ε cos(x) the zeroth Fourier modeFf2(0) = 1 causes the relative error on the first Ff2(1) = ε
2 to
be constant in expectation only for Np ∼ 1ε2
, thus depending on theamplitude of the perturbation. Removing the zeroth Fourier modewith a control variate h2(x) = 1 the relative error to be of order
12√Np
.
• A one dimensional plasma density with a small spatial perturbation
of a Maxwellian background f3(x, v) := (1 + ε cos(2πx)) 1√2πe−
v2
2 .
Taking the zeroth spatial Fourier mode∫ 1
0 f3(x, v) dx =
1 · 1√2πe−
v2
2 =: h3(x, v), here the Maxwellian background as Control
Variate yields the same variance reduction as in the previous case.
• Even for a perturbed Maxwellian velocity distributionthe standard Maxwellian control variate is good choice
f4(x, v) := (1 + εx cos(2πx))(1+εv cos(6πv))√
2πe−
v2
2 .
The Aliasing Problem
• Finite Element PIC codes based on B-Splines suffer from aliasing,which means that even under Fourier filtering high frequencies appearin a low frequency interval.
• By Fourier transform get the high frequency behavior of m-th degreeB-Spline Sm
F(Sm)(ω) = sinc(ω
2
)m+1=
(2sin
(ω2
)ω
)m+1
∈ O(
1
ωm+1
)• Estimating the Fourier modes directly - as in PIF - yields no aliasing of
the energy of other frequencies, which allows calculating the error dueto aliasing for a Fourier filtered PIC simulation. By increasing theB-Spline degree aliasing is suppressed and the PIC energy estimateconverges to the PIF estimate.
• Bump-on-tail instability [4] , Nh = 32, Np = 106, filter = 1 :10, rk2s
Discretisation of the Cylinder
• Use Fourier modes (PIF) in poloidal and toroidal direction and B-Splines (PIC) for the radial component.
• Mesh grading in the radial component with knots rk and gradingparameter αr > 0. αr = 1 uniform spacing, αr = 0.5 equiareal cells,αr = 2 resolving singularity.
rk := Rmax
(k
Nr
)αr, k = 1, . . . , Nr
Summary
• Particle in Fourier allows both momentum and energy conserving par-ticle simulations.
• Due to its slim structure PIF eases the study of stochastic methodsin theory and implementation.
• The aliasing problems in PIC codes, which are resolved by PIF, canbe studied.
• With PIF turbulent transport simulations in the poloidal plane andthe cylinder have been conducted and will be extended to the fulltorus.
Guiding Center Model (2D)
• A guiding center type equation on the polar plane Ω [2]∂tρ + (∇Φ)y∂xρ− (∇Φ)x∂yρ = 0 on Ω× [0,∞)
−4Φ = γρ
Φ(x, y) = 0 on ∂Ω
• Diocotron Instability r− = 4, r+ = 5, rmax = 10, ε = 10−2, γ = −1.
ρ(t = 0, r, θ) =
1 + ε cos(lθ) for r− ≤ r ≤ r+
0 else.
Drift kinetic model (3D+1V)
• Drift kinetic ions with adiabatic electrons in a cylindrical domain[2,3].
∂f∂t + ~vGC · ∇⊥f + v‖
∂f∂φ +
dv‖dt ·
∂f∂v‖
= 0
−∇⊥ · (n0(r)∇⊥Φ) +n0(r)Te(r)
(Φ− Φ
)=∫f dv‖ − n0(r)
Φ(r, θ, t) := 1Lϕ
∫ Lϕ0 Φ(r, θ, φ, t)dϕ
• Ion Temperature Gradient instability in the linear phase
full f
δf (with local Maxwellian as control variate)
References
References
[1] A. Y. Aydemir. “A unified Monte Carlo interpretation of particle simulationsand applications to non-neutral plasmas”. In: Physics of Plasmas (1994-present)1.4 (1994), pp. 822–831.
[2] N. Crouseilles et al. “Semi-Lagrangian simulations on polar grids: from dio-cotron instability to ITG turbulence”. Feb. 2014. url: https://hal.archives-ouvertes.fr/hal-00977342.
[3] V. Grandgirard et al. “A drift-kinetic Semi-Lagrangian 4D code for ion turbu-lence simulation”. In: Journal of Computational Physics 217.2 (2006), pp. 395–423. issn: 0021-9991. doi: 10.1016/j.jcp.2006.01.023. url: http://www.sciencedirect.com/science/article/pii/S0021999106000155.
[4] T. Nakamura and T. Yabe. “Cubic interpolated propagation scheme for solvingthe hyper-dimensional vlasov—poisson equation in phase space”. In: ComputerPhysics Communications 120.2 (1999), pp. 122–154.
Email: [email protected] Advisory Board Meeting, 4th September 2015, Greifswald
