A high-order accurate Particle-In-Cell method for Vlasov ......2016/11/10 · A high-order accurate...

A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time integrations

Andrew Myers [email protected] Applied Numerical Algorithms Group, Computational Research Division with Phillip Colella, Brian Van Straalen, Bei Wang, Boris Lo, Victor Minden

Advanced Modeling Seminar Series NASA Ames Research Center November10th, 2016

https://arxiv.org/abs/1602.00747

• Background and motivation

• Description of our numerical method

• Implementation using the Chombo framework

• Numerical results

• Summary and future research directions

A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time

2

Talk Outline







3

Talk Outline

@f

@t+ v · @f

@x�E · @f

@v= 0

r2� = �⇢ E = �r�


The Vlasov-Poisson Equations4

• The Vlasov-Poisson equations describe the evolution of a charged,

collisionless fluid in phase space.

• The electric field is obtained by solving Poisson’s equation.

• Applicable as a simplified model to plasma physics (space plasmas,

accelerator modeling, controlled fusion) and to cosmology.


Eulerian Methods5

• Nonlinear advection equation in a high-dimensional space, can be

solved by Eulerian techniques.

• Advantages:

– Strong body of theory on finite volume methods

– Do not suffer from “noise”

– High order methods exist (Vogman + 2014)

• Disadvantages:

– High cost of grids in high-dimensional (4D, 5D, or 6D) spaces

• Discretize system with set of Lagrangian interpolating points,

• Reduces problem to system of ODEs for particle trajectories:

P

(x p(t), vp(t))

f(x , v , tini) ⇡X

p2Pqp�

�x � x

ip

��v � v

ip

�

dqpdt

= 0

dx p

dt= vp

dvp

dt= �Ep


6

Particle Methods

• Can reconstruct distribution at later times from

vp

Ep

x p(t)


Particle Methods7

• Variety of methods - mainly differ in how the force is computed

• PIC methods, in which the force solve is performed on an intermediate

grid, are particularly popular.

• Advantages:

– Mathematically simpler. Reduces a PDE to a set of coupled

ODEs.

– Naturally adaptive - the particles go where more resolution is

needed.

– Usually lower computational cost


Disadvantages of Particle Methods8

• Less

mathematical

guidance

• Particle noise

prevents

convergence

• Usually limited

to 2nd order

accuracy

Wang+2011


The goal of this talk9

• Describe a particle method (PIC) that attempts to address these

downsides.

• Show how to all the PIC stages at 4th-order

– Heart of this is really the interpolation kernels.

• Describe a high-order remapping procedure to ameliorate particle

noise while maintaining 4th order accuracy







10

Talk Outline

• Field solve: force is computed on the mesh by e.g.

solving Poisson’s Equation w/ 2nd order finite

differences.

• Interpolation: Force is interpolated back to particle

positions using same kernel.

• Particle Push: Particle positions and velocities are

updated. 2nd-order leapfrog.

• Deposition: Particle charges are deposited onto mesh:

⇢

n+1i

=X

p

⇣qp

�x

⌘W

x i � x

n+1p

�x

!


11

Overview of a Particle-in-Cell time step

2nd order: Piecewise linear, Cloud-in-Cell interpolation

Start

Deposition

Field Solve

Interpolation

Particle Push

• Given solution at a set of points,

reconstruct in between

• Functions W(x) need to be even,

normalized, have compact support

⇢(x) ⇡X

j

W

✓xj � x

�x

◆⇢j

⇢0⇢1

x0 x1 ...

...


High-order in space (interpolation kernels)12

M2(x) =

⇢1� |x|, 0 |x| 1,

0 otherwise.

M3(x) =

8<

:

34 � |x|2, 0 |x| 1/2,

12

�32 � |x|

�2, 1/2 |x| 3/2,

0 otherwise.

M1(x) =

⇢1, 0 |x| 1/2,

0 otherwise.



• Commonly taken to be one of the B-spline functions. Limited to 2nd Order.

NGP CIC TSC

• Error analysis involves looking at the Fourier transform of ,

• Can show that must have a zero of order n at AND must

have zeros of order n at to be O(n).

• For the Basis spline of order n,

• Fulfills the second requirement, but zero at k = 0 is only order 2.

• Hence, all the basis splines interpolate with at best 2nd order accuracy

• They do, however, have increasing degrees of smoothness which may be

desirable when the particles are highly disordered.

• See Schoenberg 1975, Monaghan 1985 for more details.

W (k)� 1

W (k)

k = 2⇡, 4⇡, ...

k = 0

W (k) =

✓sin (k/2)

k/2

◆n



W (x)

W (k)

W (x)



• However, it is possible to design a polynomial with the appropriate zeros in

Fourier space and transform back into real space to obtain a with the

desired properties.

• For details, see Lo, Minden, and Colella 2016.


16

B a0 = 1

˜Wq

(k)

Wq

(x) =

q/2�1X

p=0

a2p

(�1)

pM (2p)q

(x),

B[�q/2, q/2]

Wq

q = 4, 6

W4(x) =

8><

>:

|x|32 � |x|2 � |x|

2 + 1, |x| 2 [0, 1],

� |x|36 + |x|2 � 11|x|

6 + 1, |x| 2 [1, 2],

0,

W6(x) =

8>>>><

>>>>:

� |x|512 +

|x|44 +

5|x|312 � 5|x|2

4 � |x|3 + 1, |x| 2 [0, 1],

|x|524 � 3|x|4

8 +

25|x|324 � 5|x|2

8 � 13|x|12 + 1, |x| 2 [1, 2],

� |x|5120 +

|x|48 � 17|x|3

24 +

15|x|28 � 137|x|

60 + 1, |x| 2 [2, 3],

0,

kk = 4

qq

[�q/2, q/2]

• 4-point stencil

• Reproduces cubic

functions exactly

• Continuous with

discontinuous first

derivative

High-order in space (interpolation kernels)

�DX

d=1

��i+2ed + 16�i+ed � 30�i + 16�i�ed � �i+2ed

12�x

2 = ⇢i

Edi = ��i+2ed + 8�i+ed � 8�i�ed + �i+2ed

12�x


17

High-order in space (field solve)

• Replace 2nd-order finite difference approximation with 4th-order, centered

differences:

r2� = �⇢

• Resulting system can be solved with a variety of methods (we used

geometric multigrid).

E = �r�


18

High-order in time (RK4)

1 The Method

The goal is to solve the following system of equations for the particle posi-

tions and velocities x and v given the force F :

x = v

v = F (t, x). (1)

The standard, fourth-order Runge-Kutta method applied to this system

gives, for the special case of a force that does not depend on v:

x

n+1= x

n+ v

n�t +

1

6

(k1 + 2k2) �t

2

v

n+1= v

n+

1

6

(k

n1 + 4k2 + k3) �t, (2)

where

k1 = F (t

n, x

n)

k2 = F

✓t

n+

1

2

�t, x

n+

1

2

v

n�t +

1

8

k1�t

2

◆

k3 = F

✓t

n+ �t, x

n+ v

n�t +

1

2

k2�t

2

◆. (3)

Note the sequential nature of this algorithm; k1 must be computed before

k2, which must be computed before k3. An alternative is to extrapolate the

forces from the previous time step. For example, the forces at the stages

corresponding to times t

n � �t, t

n � 12�t, and t

nwere k

n1 , k

n2 , and k

n3 .

Defining

⌧ =

t� (t

n ��t)

�t

, (4)

a 2nd order interpolating polynomial that passes through the required points

is

˜

f(⌧) = (2k

n1 � 4k

n2 + 2k

n3 ) ⌧

2+ (�3k

n1 + 4k

n2 � k

n3 ) ⌧ + k

n1 . (5)

For ⌧ = 0, 1/2, and 1, we recover k

n1 , k

n2 , and k

n3 , respectively. Extrapolated

forward to ⌧ = 1, 3/2, and 2 (t = t

n, t

n+ 1/2�t, and t

n+ �t), we find:

˜

f(1) = k

n3

˜

f

✓3

2

◆= k

n1 � 3k

n2 + 3k

n3

˜

f(2) = 3k

n1 � 8k

n2 + 6k

n3 . (6)

1

1 The Method

The goal is to solve the following system of equations for the particle posi-

tions and velocities x and v given the force F :

x = v

v = F (t, x). (1)

The standard, fourth-order Runge-Kutta method applied to this system

gives, for the special case of a force that does not depend on v:

x

n+1= x

n+ v

n�t +

1

6

(k1 + 2k2) �t

2

v

n+1= v

n+

1

6

(k

n1 + 4k2 + k3) �t, (2)

where

k1 = F (t

n, x

n)

k2 = F

✓t

n+

1

2

�t, x

n+

1

2

v

n�t +

1

8

k1�t

2

◆

k3 = F

✓t

n+ �t, x

n+ v

n�t +

1

2

k2�t

2

◆. (3)

Note the sequential nature of this algorithm; k1 must be computed before

k2, which must be computed before k3. An alternative is to extrapolate the

forces from the previous time step. For example, the forces at the stages

corresponding to times t

n � �t, t

n � 12�t, and t

nwere k

n1 , k

n2 , and k

n3 .

Defining

⌧ =

t� (t

n ��t)

�t

, (4)

a 2nd order interpolating polynomial that passes through the required points

is

˜

f(⌧) = (2k

n1 � 4k

n2 + 2k

n3 ) ⌧

2+ (�3k

n1 + 4k

n2 � k

n3 ) ⌧ + k

n1 . (5)

For ⌧ = 0, 1/2, and 1, we recover k

n1 , k

n2 , and k

n3 , respectively. Extrapolated

forward to ⌧ = 1, 3/2, and 2 (t = t

n, t

n+ 1/2�t, and t

n+ �t), we find:

˜

f(1) = k

n3

˜

f

✓3

2

◆= k

n1 � 3k

n2 + 3k

n3

˜

f(2) = 3k

n1 � 8k

n2 + 6k

n3 . (6)

1

• Replace leapfrog with 4th-Order Runge-Kutta method

• Requires 3 force solves per time step (velocity independent force), instead of 1

• Self-starting. Gives up on symplectic property.

eE(x , t) / exp (at)

19

Particle Remapping

• In PIC convergence theory, stability error for field contains

exponential term:

• Periodically restart problem with new particles

with interpolated weights

(Wang+2011)

Before remap After remap

Particles with tiny

masses are discarded

20

Particle Remapping

B a0 = 1

˜Wq

(k)

Wq

(x) =

q/2�1X

p=0

a2p

(�1)

pM (2p)q

(x),

B[�q/2, q/2]

Wq

q = 4, 6

W4(x) =

8><

>:

|x|32 � |x|2 � |x|

2 + 1, |x| 2 [0, 1],

� |x|36 + |x|2 � 11|x|

6 + 1, |x| 2 [1, 2],

0,

W6(x) =

8>>>><

>>>>:

� |x|512 +

|x|44 +

5|x|312 � 5|x|2

4 � |x|3 + 1, |x| 2 [0, 1],

|x|524 � 3|x|4

8 +

25|x|324 � 5|x|2

8 � 13|x|12 + 1, |x| 2 [1, 2],

� |x|5120 +

|x|48 � 17|x|3

24 +

15|x|28 � 137|x|

60 + 1, |x| 2 [2, 3],

0,

kk = 4

qq

[�q/2, q/2]

• One order of accuracy is

lost during the remap step

• High-order deposition

• 6-point, 6th-order stencil

q⇤p

=X

p

qp

W 6

✓x

i

� x

p

hx

◆W 6

✓v

j

� v

p

hv

◆

21

Particle Remapping

• Particles are laid on at the cell centers of a (possibly AMR) Cartesian

grid in phase space. The new weights are then computed as:

• Particles with small weights are discarded.

• Can be applied every few time steps, or just a few times per calculation

• Can be thought of as Semi-Lagrangian

• Compare the Forward Semi-Lagrangian technique (Crouseilles+2008)

Z

RD

(x� y)`W (x� y)dy

22

Particle Remapping

• Higher order interpolation functions

are not positivity preserving

• Can show that the 2nd moment of the

interpolating function must vanish for

the kernel to have better than 2nd

order accuracy.

Positivity not guaranteed

Z

RD

(x� y)`W (x� y)dy

23

Particle Remapping

• Higher order interpolation functions

are not positivity preserving

• Use a redistribution procedure to

maintain positivity of the distribution

function

Positivity not guaranteed

�fi = min(0, fni )

⇠i+l = max(0, fni+l )

fn+1

i+l = fni+l +

⇠i+lPneighbors

k 6=0

⇠i+k

�fi

l

Redistribution procedure24

• 2 or 3 iterations is usually sufficient

neighboring cells

25

Particle Remapping AMR example

• Example distribution

function from a remapping

stage, using 4 AMR levels

• Cosmological dark matter

test problem (Myers+2015)

Remapping on Cosmology Calculation26

Remapped Not remapped

http://adsabs.harvard.edu/abs/2016ApJ...816...56M







27

Talk Outline

• Chombo - A set of tools for numerically solving PDEs • Especially useful for AMR applications • Mixture of C++ and FORTRAN • Uses a Box-based, SPMD approach to parallelism • Provides distributed data containers, including (in the next release)

for particle data • Our method was implemented using these tools, running NERSC’s

Edison machine + my laptop. • Code, tests, and analysis scripts are all here:


28

Implementation

https://bitbucket.org/atmyers/4thorderpic







29

Talk Outline


Linear Landau Damping

• Exponentially damped oscillation of a space charge wave • Wave-particle interactions cause the electric field wave to lose

energy to the particles, damping its amplitude • Very commonly used test problem for PIC codes

– Analytic solution for the decay rate – Long-time evolution is challenging for classical PIC methods

30

• Gaussian velocity distribution

• Small (linear) perturbation to the charge density (physical space)

• Track the electric field as a function of time

f(x, v, t = 0) =

1p2⇡

exp

��v

2

/2

�(1 + ↵ cos (kx))

(x, v) = [0, L = 2⇡/k]⇥ [�v

max

, v

max

]


Linear Landau Damping (initial conditions)31

k = 0.5

↵ = 0.01


Linear Landau Damping (movie)32


Linear Landau Damping (results)33

• Black shows simulation result, red shows damping rate expected from linear theory.

• Remapping is needed to track the analytical damping rate for late times.

• Recurrence effect seen in all simulations past t = 80, a consequence of finite cell spacing in the velocity dimension

• All discrete elements of the problem, including , reduced by factor of 2.

• Ratio of remains fixed. �x/h

x


34

Convergence studies for PIC simulations

�t

• We then use Richardson extrapolation to compute the error. • The error in the electric field is then defined as:

eh = |Eh �E2h|

q = log2

✓||e2h||||eh||

◆


35

Convergence studies for PIC simulations

• And the order is computed as:


Linear Landau Damping (convergence results)36


Linear Landau Damping (results)37

Remapping every 5 steps gives the best results, but even every 100 helps

f(x, v, t = 0) =

1p2⇡

exp

��v

2

/2

�(1 + ↵ cos (kx))

(x, v) = [0, L = 2⇡/k]⇥ [�v

max

, v

max

]


Nonlinear Landau Damping (initial conditions)38

k = 0.5↵ = 0.5


Nonlinear Landau Damping (movie)39


Nonlinear Landau Damping (results)40


Nonlinear Landau Damping (convergence)41


Nonlinear Landau Damping (Phase Diagrams)42

• Initial distribution function corresponds to two particle streams with opposing velocities:

f(x, v, t = 0) =

1p2⇡

v

2

exp

��v

2

/2

�(1 + ↵ cos (kx))

(x, v) = [0, L = 2⇡/k]⇥ [�v

max

, v

max

]


Two-Stream Instability43

↵ = 0.01k = 0.5

“Inverse” of Landau Damping



Remapping 10 times (out of ~1000)


Two-Stream Instability (convergence results)47


49

http://yt-project.org/







50

Talk Outline


Summary and Future Research

• High-order method is more accurate in regions where the flow is smooth

• Not more necessarily more accurate for thin or unresolved phase-space features in the nonlinear regime.

• Remapping is necessary to control particle noise over long time evolutions

• Involves some degree of retreat from pure Lagrangian. AMR helps to an extent.Trade-offs need to be explored.

• Future research involves performing the remap selectively rather than globally.

• Coupled with high-order Maxwell solver for electromagnetic PIC (Sven Chilton)

51

Thanks for listening!

Andrew Myers [email protected] Applied Numerical Algorithms Group, Computational Research Division with Phillip Colella, Brian Van Straalen, Bei Wang, Boris Lo, Victor Minden

Advanced Modeling Seminar Series NASA Ames Research Center November 10th, 2016

https://arxiv.org/abs/1602.00747

A high-order accurate Particle-In-Cell method for Vlasov ......2016/11/10 · A high-order accurate...

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