a particle-in-cell method with adaptive phase-space...
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A particle-in-cell method with adaptivephase-space remapping for kinetic plasmas
Bei Wang1 Greg Miller2 Phil Colella3
1Princeton Institute of Computational Science and Engineering
Princeton University
2Department of Chemical Engineering and Materials Science
University of California at Davis
3Applied Numerical Algorithm Group
Lawrence Berkeley National Laboratory
Mathematical and Computer Science Approaches to High Energy Density PhysicsWorkshop, IPAM
May 10, 2011
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Outline
BackgroundVlasov equationNumerical methods for the Vlasov equation
Particle MethodsThe particle in cell methodWhat is particle noise?
Noise Reduction: Phase-space RemappingConservative, high order and positive remappingMesh refinement
Numerical results
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The Vlasov-Poisson equation
We consider a collisionless electrostatic plasma with two species,electron and ion, described by the Vlasov-Poisson equation
Vlasov equation
∂fe∂t
+ v · ∇xfe + F · ∇vfe = 0
F = − qeme
E
Poisson equation
ρ = 1− qe
∫Rn
fedv
∆φ = −ρ E = −∇φ
fe(x, v, t): the electron distribution function in phase spaceF(x, t): Lorentz force (electrostatic case)E(x, t): the self-consistent electrostatic field
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Lagrangian description of the system
The distribution function f is conserved along the characteristics.At time s and t
f (X(t),V(t), t) = f (X(t0),V(t0), t0)
where (X(t),V(t)) is the characteristics of the Vlasov equation:
dX
dt= V(t)
dV
dt= F(t)
where X(t0) = x0 and V(t0) = v0
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Review of numerical methods: grid methods
1. Grid-based methods include spectral methods (Knorr 63,Armstrong 69, Flimas-Farrell 94), semi-Lagrangian methods(Cheng-Knorr 76, Sonnendrucker 99, Nakamura-Yabe 99),finite volume methods (Fijalkow 99, Filbet 01, Colella 11) andfinite element methods (Zaki 88, Kilimas-Farrell 94).
2. They have drawn much attention in the past decade thanks toincreasing processing power.
I Advantage: Smooth representation of fI Disadvantage: High dimensions (up to 6) −→ high
computational cost (specifically memory)
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Review of numerical methods: particle methods
Particle methods,e.g.,the PIC method,are widely used and arepreferred for high dimensions.Advantages:
I Naturally adaptive, since particles only occupy spaces wherethe distribution function is not zero
I Simpler to implement, in particular in high dimensions
Disadvantages:
I particle noise −→ difficulties to get precise results in somecases, for example, in simulating the problems with largedynamic ranges
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Particle methodsI In particle methods, we approximate the distribution function by a collection of
finite-size particles
f (x , v , t) ≈∑k
qkδεx (x − Xk (t))δεv (v − Vk (t))
qk = f (xi , vi , 0)hxhv
where Xk (0) = xi , Vk (t) = vi .∫ ∞−∞
δεx (y)dy = 1
δεx (y) =1
εxu(
y
εx)
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u(y)
y
u1u0
I At each time step, particles are transported along trajectories described by theequation of motion
dqk
dt= 0
dXk
dt= Vk (t)
dVk
dt= F k (t)
where Xk (0) = xk and Vk (0) = vk .
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The PIC method
I Charge assignment:
ρ(xj , tn) =
∑k
qk
εxu1(
xj − Xk (tn)
εx)
where j is the grid index.
I Field solver: i.e., FFTs or multigrid methods
φj−1 − 2φj + φj+1
εx 2= ρj
Ej =φj−1 − φj+1
2εx
I Force interpolation:
E(Xk , tn) =
∑j
Eju1(xj − Xk (tn)
εx)
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What is particle noise?
Particle noise: the numerical error introduced when evaluating themoments of the distribution function using particles in phase spaceand the particle disorder induced by the numerical errorError analysis:
I Monte Carlo estimate (Aydemir 93)
error ∝ σ√N
where σ is the standard deviation depends on the particle sampling and the
distribution function.
I The approach in vortex methods: consistency error + stabilityerror (Cottet-Raviart 84):
error ∝ consistency error × (exp(at)− 1)
where a is a physical parameter.
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Error Analysis of the PIC method for the VP system
The charge density error is
|ρ(x , t)− ρ(x , t)| = |ρ(x , t)−∑k
qkδεx (x − Xk (t))|
≤∣∣∣∣ρ(x , t)−
∫Rρ(y , t)δεx (x − y)dy
∣∣∣∣︸ ︷︷ ︸moment error:em(x,t)∝ε2
x
+
∣∣∣∣∣∫Rρ(y , t)δεx (x − y)dy −
∑k
qkδεx (x − Xk (t))
∣∣∣∣∣︸ ︷︷ ︸discretization error:ed (x,t)∝εx 2( hx
εx)2
+
∣∣∣∣∣∑k
qkδεx (x − Xk (t))−∑k
qkδεx (x − Xk (t))
∣∣∣∣∣︸ ︷︷ ︸stability error:es (x,t)∝ 1
εxmaxk |Xk−Xk |
|E(x , t)− E(x , t)| ∝(εx
2 + εx2
(hx
εx
)2
+ maxk|Xk (t)− Xk (t)|
)
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By following Cottet and Raviart (84):
(Xk − Xk )(t) =
∫ t
0(Vk − Vk )(t′)dt′
and
(Vk − Vk )(t) = −∫ t
0
(E(Xk , t
′)− E(Xk , t′))dt′
= −∫ t
0
((E − E)(Xk , t
′) +(E(Xk , t
′)− E(Xk , t′)))
dt′
= −∫ t
0
((E − E)(Xk , t
′) +∂E
∂x(Xk − Xk )(t′)
)dt′
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By using a variation on Gronwall’s inequality, we get
maxk
(|Xk (t)− Xk (t)|+ |Vk (t)− Vk (t)|+∥∥∥E − E(·, t)
∥∥∥L∞
)
≤ C1
εx 2 + εx2(
hx
εx)
2
︸ ︷︷ ︸ec (x,t)
+
(εx
2 + εx2(
hx
εx)2
)(exp(at)− 1)︸ ︷︷ ︸
es (x,t)
where a is max(1,
∥∥∥ ∂E∂x
(·, t)∥∥∥L∞
), but not εx and hx .
Requirements for Convergence
I Particle overlapping: hxεx≤ 1
I Particle regularization: control the exponential-like term
ref: Cottet-Raviart 84 and Wang,Miller,Colella 11
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Options if we want to reduce particle noise
I Perturbative methods, such as the δf method(Kotschenreuther 88, Dimits-Lee 93, Parker-Lee 93):discretize the perturbation with respect to a (local)Maxwellian in velocity space using particles
⇒ reduce σ ( error ∝ σ√N
)
I Our approach: remapping: remap the distorted chargedistribution on regularized grid(s) in phase space and thencreate a new set of particle charges from the grids withregularized distribution
⇒ reduce exponential term
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Conservative remapping on phase space (x, v)
Particle charges are remapped (interpolated) to a grid in phasespace
qi = qku(xk − xix
hx)u(
vk − vivhv
)
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u(y)
y
u1u0
Total charge and momentum are conserved if the interpolationfunction u satisfies ∑
i∈Z u( x−xih ) = 1∑i∈Z xiu( x−xih ) = x
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High order interpolation
A modified B-spline by Monaghan (85)
u2(y) =
1− 5|y |2
2 + 3|y |32 0 ≤ |y | ≤ 1
12 (2− |y |)2(1− |y |) 1 < |y | ≤ 2
0 otherwise
-0.2
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u(y)
y
u2
u2 can approximate a quadratic function exactly (error is O(h3)).Moreover, its first and second order derivatives are continuous.
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Positivity for high order interpolation functionsAlgorithm: Global mass redistribution based on flux correctedtransport (Zalesak 78)Treat interpolation as advection
f n+1i = f ni +5 · F
f n+1i =
∑k
qk1
hxhvu1|2(
∣∣∣∣xi − xkh
∣∣∣∣) f ni =∑k
qk1
hxhvu0(
∣∣∣∣ xi − xkh
∣∣∣∣)
-0.2
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
y
x
u1u2u0
Obtain the flux by solving a Poisson equation 5 · F = f n+1i − f ni
Define a low order flux Flo and a high order flux Fhi as being usinga low order u1 and a high order u2 interpolation.Correct the low interpolation value by using the idea of FCT
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Positivity for high order interpolation functions
Algorithm: Local mass redistribution by Chern-Colella (87)We redistribute the undershoot of cell i
δfi = min(0, f ni )
to neighboring cells in proportion to theircapacity ξ
ξi+` = max(0, f ni+`)
The distribution function is conserved,which fixes the constant of proportionality
f n+1i+` = f ni+` +
ξi+`
neighbors∑`′ 6=0
ξi+`′
δfi
Cell with Negative Charge
Neighboring Cells
`
i
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Mesh refinement
Motivation: Remapping creates too many small strength particlesat the edge of the distribution function.Algorithm: Interpolate as in uniform grid first, then transfer thecharge from invalid cells to valid cells
Ω`+1c
Ω`c
Particle is at the coarser level side
Ω`+1c
Ω`c
Particle is at the finer level side
The valid deposit cells: filled circles The invalid cell: open circles
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Introduce collision term
1. Remapping provides an opportunity to integrate collisionmodels with a grid-based solver.
2. Example: simplified Fokker-Planck equation suggested byRathmann and Denavit (75)
(∂f
∂t)c = ∇v · [νvf + D∇v (νf )]
where (ν,D) are constants.
I Discretize it using a finite volume discretization and a secondorder L0 stable, implicit scheme
I Solve the matrix system using multigrid methodI Couple it with the Vlasov equation with operator splitting
∂f
∂t+ v ·
∂f
∂x− E ·
∂f
∂v= (
∂f
∂t)c
N+1t
N+1/2N
R R+C
PIC
R
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1D Vlasov-Poisson: linear Landau damping
The initial distribution of linear Landau damping is
f0(x , v) =1√
2πexp(−v2/2)(1 + α cos(kx))
where α = 0.01, k = 0.5, L = 2π/k.
The evolution of the amplitude of the electric field: exponential decay with rateγ = −0.1533
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
t
γ=-0.1533no remapping, hx=L/32, hv=hmax/32
w/o remapping
particle number: 960
CPU times: 3.99 seconds
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
t
γ=-0.1533with remapping, hx=L/32, hv=hmax/16
w/ remapping
particle number: 960-1,539
CPU times: 5.21 seconds
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1D Vlasov-Poisson: linear Landau damping
1e-05
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
t
γ=-0.1533no remapping, hx=L/128, hv=hmax/256
w/o remapping
particle number: 60,416
CPU times: 141.60 seconds
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
t
γ=-0.1533with remapping, hx=L/32, hv=hmax/16
w/ remapping
particle number: 1,539
CPU times: 5.21 seconds
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1D Vlasov-Poisson: the two stream instability
The initial distribution of the two stream instability is
f0(x , v) =1√
2πv2exp(−v2/2)(1 + α cos(kx))
where α = 0.05, k = 0.5, L = 2π/k.
The evolution of f(x,v,t)
w/o remapping, particle number: 18,360 w/ remapping, particle number: 18,360-24,242
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1D Vlasov-Poisson: the two stream instability
Comparison of f (x , v , t) at the same instant of time t = 20
w/o remapping maxc = 1.740 vs. maxe = 0.3 w/ remapping maxc = 0.3082 vs. maxe = 0.3
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1D Vlasov-Poisson: the two stream instability
E field Errors and convergence rates
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
erro
r
t
hx=L/64, hv=vmax/128hx=L/128, hv=vmax/256hx=L/256, hv=vmax/512
-1
0
1
2
3
0 5 10 15 20 25 30
conv
erge
nce
rate
t
hx=L/128, hv=vmax/256hx=L/256, hv=vmax/512
w/o remapping
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30
erro
r
t
hx=L/64, hv=vmax/128hx=L/128, hv=vmax/256hx=L/256, hv=vmax/512
-1
0
1
2
3
0 5 10 15 20 25 30
conv
erge
nce
rate
t
hx=L/128, hv=vmax/256hx=L/256, hv=vmax/512
w/ remapping
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1D Vlasov-Poisson-Fokker-Planck: the two streaminstability
The evolution of f(x,v,t)
Simulation w/o (left) and w/ (right) collision at t=30
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2D Vlasov-Poisson:linear Landau damping
The initial distribution of linear Landau damping in 2D is
f0(x , y , vx , vy ) =1
2πexp(−(v2
x + v2y )/2)(1 + α cos(kx) cos(ky))
where α = 0.05, k = 0.5, L = 2π/k.
The evolution of the electric energy ξe : exponential decay with constant rate γ
-20
-18
-16
-14
-12
-10
-8
-6
0 2 4 6 8 10 12 14 16
log(
ξ e)
t
γwithout remapping, hx=L/32, hvx
=hmax/16
w/o remapping, particle number: 1,228,800
-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
0 2 4 6 8 10 12 14 16
log(
ξ e)
t
γwith remapping, hx=L/32, hvx
=hmax/16
w/ remapping, particle number: 1,228,800-1,835,008
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2D Vlasov-Poisson: the two stream instability
The initial distribution for the two stream instability in 2D is
f0(x , y , vx , vy ) =1
12πexp(−(v2
x + v2y )/2)(1 + α cos(kxx))(1 + 5v2
x )
where α = 0.05, kx = 0.5, L = 2π/k.
Comparison of projected distribution function in (x , vx ) at time t = 20
w/o remapping, particle number: 14,408,192 w/ remapping particle number: 14,408,192-22,184,217
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Code implementation
The parallel and multidimensional Vlasov-Poisson solver isimplemented using Chombo framework, a C/C++ and FORTRANlibrary for the solutions of partial differential equations on ahierarchy of block-structured grid with finite difference methodsdeveloped in the ANAG group of LBNL.
I The physical domain is decomposed into patches
I Each patch is assigned to a processor
I Particles are assigned to a processor according to theirphysical position
I Particles are transferred between patches using MPI
Continuous work in ANAG: scalable multidimensional particle incell code with remapping
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Future Work
I Adaptivity on creating a hierarchy of grids for remapping
I Apply the method to magnetic fusion plasmas, e.g.,gyrokinetic particle in cell method
I Parallel scalability
I GPU acceleration of remapping
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Thank you!
Questions?