AC274:Particles in Cell

(phase-space fluids)

Systems with soft (long-range) interations:

Grazing collisions (small deflections)

Each particle feels a collective soft interaction

Plasmas, AstrophysicsBiological fluids,Electrophoresis,

Phase-space fluids

Particle-Particle: hard-core

Field-Particle: smooth

Particle: mesh-free

Field: mesh bound

Electro-biofluids: Ion Channels

Vlasov-Poisson-Boltzmann

Superparticles

Smooth force:

Length scales and resolution

1. Charge assign to the grid (P2G)

2. Solve field eq.s on the grid (GG)

3. Compute forces on the particles (G2P)

4. Move particles (PP)

PIC: solution procedure

G-P duality

Transfer operators: Weight and Green

P2G:

G2P:

Charge

PotentialForce

PIC algorithm

2.GG: Solve Poisson:

1. GG: Charge assign:

Vlasov-Poisson

3. GP: Force interpolation

4. PP: Move Particles

Charge assign: scatter, P2G

Solve Poisson, GG

Potential to Force (GG)

Force transfer G2P

Particle Mover

Charge Assign: Locate

Grid NE

SE

NW

SW

P=(x,y)

Locating particles in arbitrary grids

Charge Assign: NGP

2

1

3

0

P

Charge Assign: PWC

NE

SE

NW

SW

P

Charge Assign: PWlin

NE=2NW=3

p=(x,y)

SW=0 SE=1a

b

Force interp: NGP

NE

SE

NW

SW

Force interp: PWC

NE

SE

NW

SW

Force interp: PWL

2

1

3

0 a

bp

Charge collect: receive

Charge assign: NGP

8

76

5

4

3

2

1

Charge assign: PWC

8

76

5

4

3

2

1

Shape functions: 1d

Spurious Forces and Grid Independence

The self-force problem

The self-force problem

ng=2

The self-force problem

PwL: -a*(1-a)+(1-a)*a; = 0 for any a

PwC: -a*1/2+(1-a)*1/2=1/2-a; = 0 only for a=1/2

NGP: -a*1, or (1-a)*1; = 0 never

BUT: if <a>=1/2 both pwc and ngp give zero on average!

Spurious forces

Choosing the kernel

Use the same kernel

Or:Force kernel G equal or lower order than charge kernel W

Poisson solver

Poisson Solver

Spectral (if you can…)

Poisson Solver: relax

Poisson Solver: RES

Special case: Laplace

Poisson Solver: bc TBD

Thomas algorithm

Bwd sweep

Fwd sweep

Thomas 2D

ADI

Particle mover

Position Verlet

Staggered: (4th, reversible)

Q: How do we start-off?

Velocity Verlet

PIC methods scale like NP*NG instead of NP*NP

Very useful for phase-space fluids with soft interactions: plasmas, astrophysics, electro-bioflows

Modern developments:Unstructured, Moving Grids, parallel solvers, totrack STRONG inhomogeneities

P3M: Particle-Particle (dense) + Particle-Mesh

Summary

End of Lecture

Nearest grid point

i=nint(x/dx)

Phi(i)=phi(x)

CIC=PWC

Phi(i)=phi(x)/2Phi(i+1)=phi(x)/2

p=x/dx-i p>0: p<0:

Velocity Verlet

P2G: Scatter

CIC Force transfer

CIC Force transfer

G-P consistency