Review and Comparison of Particle-in-Cell andVlasov Simulation methods with application to

relativistic self-focusingJames Koga

Advanced Photon Research Center, JAERI, Kyoto-fu, 619-0215, Japan

Abstract. In this paper we present a review and comparison of Particle-in-Cell and Vlasov methodsfor plasma simulation with applications to relativistic self-focusing of high intensity laser pulses inplasmas.

INTRODUCTION

Plasma phenomena occur throughout nature and are the result of the complex natureof the collective interaction of many charged particles. Simulation of plasmas on largescale computers has become an invaluable tool in analyzing various aspects of plasmabehavior. In particular for laser plasma interaction it has become the dominant meansof explaining results of experiments using high intensity short pulse lasers. In this paperwe will be discussing two types of plasma simulation techniques. They are particle andVlasov simulation techniques.

PARTICLE SIMULATION

In the real world individual charged particles in a plasma are coupled to each other viaelectromagnetic fields (E,B). Particles are accelerated by the electromagnetic fields viathe Lorentz force equation:

- (2)dt~jm ()

where x is the position, p is the momentum, q is the charge, y is the relativistic factor,and m is the mass of the particle. In particle simulation a large number of simulationparticles are advanced by these equations.

Typically a finite differencing scheme is used to advance the particles which wasdeveloped by Boris [1], First, we rewrite equation 1 in the form:

dt m jc

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388

where u = yv. Finite differencing this equation we get:

-un~2 q ,^n 2"+2 +un~— ^ T?n j_ __Af (4)

where the superscript n refers to whole time steps and n±^ refers to fractional timesteps. This equation contains both E and B. One can eliminate E by expressing intro-ducing the following variables:

2m

where y1 = 1 + (^-) . Rewriting equation 3 we get:

where

4M+ = I «+

«z+

«7

Equation 6 represents a matrix equation which can be inverted to get:

1 „ _

(5)

(6)

(7)

(8)

, (9)

where Q2 - Q2 + Q2 + Q2, Q^ = ̂ , Qy - ̂ , and Dz

and 8 we can get wn+2. Finite differencing equation 2 we get:

Q2 -

. Using equations 5

A;(10)

where (7"+5)2 = 1 + (!L^)2 which can be used to advance particle positions.One way the electromagnetic fields used in Equation 1 can be calculated is to calculate

the contribution from other particles in the plasma via the Lienard-Wiechert fields [2]:

i = e H - P

ret

I _C

3x{(n-p)xp} (J,r) =\nxE] (11)L J re/

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where E(x,t) and B(x,t) are the electric and magnetic fields, respectively, generated bycharged particles other than the particle which feels the field. Here, ret refers to the timein the past where the trajectory of the other particle intersects with the light cone of theparticle which sees the fields, n is the unit normal vector between the particle and otherparticle's past position, (3 and y are the usual relativistic factors, and p is d$/dt which isthe usual acceleration divided by c.

From a computational point of view it can be seen that if there are N particles whichinteract via the Lienard-Wiechert fields then N2 interactions must be calculated. Re-sultingly, the amount of computation increases rapidly with particle number so only alimited number of particles can be calculated in a reasonable time even using supercom-puters.

One way of getting around the amount of computation required from direct particle-particle interaction simulations is to compute the electromagnetic fields on a finite num-ber of grids. This method is called Particle-in-Cell (PIC). Many excellent references canbe found describing this method [3, 4, 5, 6] so in this section we will only cover brieflythe essential details of the method. In the PIC method there are still particles, however,the field through which they interact is calculated using grids on which Maxwell's equa-tions are solved:

V-£ = 47tp V-B = Q (12)

VxE = --c^ V x £ = f / + I f (13)

where p refers to the charge density and /refers to the current density. The charge andcurrent density are accumulated on the grid from the particles. By using grids instead ofcalculating direct interactions the number of calculations for a N particle system goesas [6]: MlnM + bN where M is the number of grids and b is a constant. The increase incomputation only goes as roughly TV as opposed to TV2 for particle-particle simulations.This makes possible calculations of the interaction of many particles through simulation.The charged particles are coupled to eachother via the grid.

There are several ways to solve Maxwell's equation on a uniform grid. They includeFast Fourier Transforms [5] and implicit finite difference schemes [4]. We will describein more detail an explicit finite difference scheme [3] which is more suitable for imple-mentation on massively parallel computers where local solutions are optimal for speed.Rearranging Maxwell's equations where equations 12 are taken as initial conditions andfinite differencing each component of the electric field (Ex, Ey,Ez) for a two dimensionalgrid we get:

A/ A;yp n+5 _ p 2DZ. • , 1 , , 1 D7- 1

(15)

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HI -

, 1k + 2

k •

k I2

V 1 -

Ay

*~t

J,5,

^

£? B,B, J,J, s,*,

*. J.

5̂J

rBs

*. ^i

^JT

B,

4Ev

\Bt

*i .̂^r 5r

5r

J

B,

fE*

\

J* *,5X i

R

^ ,̂

5,

Y,r

N'j

R

;

FIGURE 1. The finite difference positions of the fields on a uniform two dimensional grid is shown.

• H+l

= c-B*

Ax— c-

(16)

and for the magnetic field (BX:By,Bz) we get

Ar = — c-A};

Z5v 77 n J7 nZ i_u ] fr-u ! z i ! t_u l7+2'^+2 7-2^+2

Ar Axn T? n

(17)

(18)

A? = -c(- Ax A^

In figure 1 we show the sequence of calculation for the fields on a uniform twodimensional grid. Note that the E and B fields are offset from one another by half timesteps and half a grid cell. This finite differencing scheme is stable as long as the Courantcondition is satisfied for the simulation time step Ar. In the case of two dimensions thecondition is[3]: cAf < Ax/\/2 assuming Ax = Ay where Ax and A;y are the grid sizes inthe x and y direction, respectively.

The current terms (Jx,Jy,Jz) in equations 14, 15, and 16 are calculated by accumu-lating the current contributions from the simulation particles onto the grid. By appro-priately accumulating current on the grid one can maintain charge conservation with-out having to recalculate equation 12. The technique is fairly detailed so we refer thereader to the reference [7]. Figure 2 shows the collection of current in the simplest

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(x+6x9y+6y)

FIGURE 2. The collection of current from a particle onto a uniform two dimensional grid (left) and theinterpolation of the fields on the grid to the particle (right) are shown. (x,y) is the initial particle positionand (jc + 5jc,y + 5y) is the final position

case where four cell boundaries are crossed by the particle. The currents are calcu-lated as: Jxl = JX2 = Jyi = andJy2 = + Jt-h 5&c), where 8jc and &y refer to the change in the particle position inone time step in the x and ;y directions, respectively. There are more complicated cross-ings of 7 and 10 boundaries which are described in [7].

Once the new fields have been calculated on the uniform grid, they need to beinterpolated to the particle position. This is done by an area weighting scheme [3]which is shown in figure 2: f ( x , y ) = f ( j , k ) ( l - &c)(l - &y) +f(j + l,£)&c(l - &y) +f(j,k+ l)8y(l — &c) + f ( j -f l , f c + l)5^8^c where / represents the field quantity beinginterpolated to the particle position.

In addition to the various proceedures described above additional constraints areplaced on the simulation due to numerical instabilities. One instability is the thermalinstability. If the temperature of the plasma particles is not high enough then the plasmawill numerically heat up unless the following condition is met [3]: ^ > 0.3 whereX/) = ^/kT/4nnoe2 is the Debye length, T is the temperature of the simulation particles,and no is the plasma density.

For an example of the application of the PIC method see the article in these proceed-ings dealing with the study of proton acceleration and relativistic self-focusing by thisauthor [8].

VLASOV SIMULATION

In this section we describe Vlasov simulation methods. In the previous section we talkedabout combinations of simulation particles and grids to model plasma behavior. In thecase of Vlasov simulation only grids are used to model the plasma. The advantage of

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V -max

0

V

————

Ax

——

AV

1 ————————————

V»< Y

L

-

FIGURE 3. The grid on which the Vlasov equations are solved. The grid has 2M + 1 cells in velocity Vwith indicies j = — M, — M+ 1,..., —1,0,1,...,M — 1, M and N cells in space X with indicies/= 1,2, ...,N

this method is that it is very accurate. The noise level is very low. Since we deal onlywith grids parallelization on massively parallel computers is fairly straightforward. Thedisadvantage of this technique is that large amounts of computer memory are neededand different types of numerical instabilities occur. In this section we describe numericalsolution of the Vlasov equation in one dimension using electrostatic fields:

-dt dx dv

I—/>(20)

(21)

where / is the distribution function /(jc,v,f), x is the position, v is the velocity, tis the time, and E is the electrostatic field E(x,t). The following normalization isused:Ax —)• A,£>, Ar —> co^ = ^/4nnoe2/m, v —> A^co^. Equations 20 and 21 are solvedon a uniform grid which is shown in figure 3.

Equation 20 is a hyperbolic equation so we can use the cubic interpolation splinetechnique (CIP)[9,10]. In addition to increase accuracy we use differential algebra (DA)which allows one to calculate derivatives algebraically, see [11]. This combination iscalled the DA-CIP scheme[12]. In the following section we will briefly describe thismethod. The reader is referred to [12] for further details.

The general form of the equations which we are solving can be written in the form:

(22)

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whereoc= (x,y,z),r = (rx,ry,rz),u= (ux,Uy,uz),andg(r,f,df/dr,t) is a forcing term.The equation for the advance of the derivatives can be written in the form:

Equations 20 and 21 can be expressed in Lagrange form as:

7^ = 0 % = v g = -*(*,') (24)

with the derivatives expressed as:

These equations can be expressed in a more compact form as:

(26)

where § = (*, v, /, dxf, dvf) and G = (v. — £*, 0, (dE/dx)dvf, —dxf) .To calculate the time advance of equation 26 one can Taylor series expand the equa-

tion:

(28)

where A? is the time step size. We can calculate this equation via a second order Runge-Kutta integration scheme:

3(f) + y(hi=G(q) h2 = G(q + hiAt) (30)

In the first step of the calculation we calculate ~h\ — G(q). In order to determine thiswe need to calculate the electric field E. We know that: E — — -^ where (|) is the scalarpotential so that after finite differencing we get £/ = ^JAx/"1 - In addition, we can getdxEi = ~ ~A^2+^~ where the indicies / are the same as figure 3. Equation 21 canthen be written in the form:

y*'-' = T /(*, v,r)*- 1 ^ [F^O*- 1 (31)AX,- J — oo •_ • JV;— —

•_ •7— — 7

7max ~~ 1(32)

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m

FIGURE 4. Each cell moves when q(t] is advanced.

This equation represents a tridiagonal matrix which can quickly be solved by the Thomasalgorithm. Once the potential <|) has been solved the electric field can be calculated. Inequation 31 the two dimensional cubic interpolation function was used F/y for f ( x , v )for (*,v) G ([X/,Xi+i], [Vj,Vj+l]) where

(33)

where some of the coefficients cnm are:

C2,l = dxfijAXi C2,2 = 3jcv

All the coeffiecents can be found in [12].The second step involves calculating 7*2 = G(q) where ^ = q + Ji\&t is the time

advanced q intermediate state. To calculate this we need the electric field at the ad-vanced time: E(x,t)\x=f. This requires the distribution function / at the advanced time:/(*, v, t + Af ) constructed from the intermediate states /, dxf and 3V/. When q(t) is ad-vanced in time each cell position also moves as seen in figure 4. We need to reconstructthe distribution function from these new cell positions. This can be done by using thecubic interpolation function in_equation 33_. Replacing the old values, fij^xfij^vfijwith the intermediate values, f i j , d x f i j , d v f i j in the definitions of the coefficients cnm.The interpolation function //j(jc, v) satisfies:

Fij(Xi, Vj) = fij dxFijft, Vj) = djij dvFij(Xi, Vj) = dvfij (34)dvFij(Xi+i,Vj) = dvfi+ij (35)dvFij(Xi,Vj+i) = dvfij+i (36)

3v^-+i j+i (37)Fij(Xi,Vj+i) =

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By using this function we can determine the values of the grid points within each newcell ABCD in figure 4. Let Rnm — (Xn, Vm) represent the grid point in ABCD. In order tocalculate the value at this point we use the cubic function. However, it is a function ofthe old cell positions in ABCD. This can be resolved by finding the mapping betweenthe new cell and old cell. To find the position R° = (X°, V°) in ABCD corresponding toRnm we assume a linear transformation between the two cells of the form:

(38)

j - nj = Tij(Ri+ij - Rtj) r/j+i - nj = Tij(Rij+\ - RIJ) (39)

where r/j and RIJ are the new and old grid positions, repectively, and 7/j is a lineartransformation matrix defined by equation 39. Once we know this transformation wecan write:

/B,m = ̂ (X0,V°) 3^ = ^j(X°,V°) dvf^n = ̂ Fij(X°,V0) (40)

This is done for all the grid cells to get /(jc,v,f + Af). Once this is done hi can bedetermined and used in equation 29 to get q*(t + A/) which is the time advanced grid.We repeat this whole process for each time step until the desired number of time stepsis reached.

COMPARISON AND CONCLUSION

Figure 5 shows results for PIC (left) and Vlasov (right) simulations with initial condi-tions at the top and final results at the bottom. The simulation run is for the two streaminstability where initially oppositly flowing electron beams are unstable and merge toform a vortex in x-v phase space. The parameters of each simulation are somewhat closeto eachother. It can be seen that there are fluctuations in the distribution function forthe PIC simulation whereas in the Vlasov simulation there is none. Each simulationconverges to a single vortex. In the case of the Vlasov simulation the distribution is un-changing after some time. However, the PIC simulation is still evolving in time. It willbe of further study to determine which type of simulation is closer to reality over longtime scales and over what time scales each type of simulation can be useful.

ACKNOWLEDGMENTS

I especially would like to acknowledge Takayuki Utsumi for his development of thetechniques described in the section on Vlasov simulation. I would like to thank KazuhisaNakajima for inviting me to give a review talk concerning plasma simulation and thestudents who attended my talk asking many thought provoking questions.

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FIGURE 5. Comparison of PIC results (left) and the Vlasov results (right) are shown for the two-streaminstability where the initial condtions are at the top and the final states are at the bottom.

REFERENCES

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Company, New York, 1985.4. Hockney, R. W., and Eastwood, J. W., Computer Simulation Using Particles, Adam Hilger, Bristol,

1988.5. Dawson, J. M., Rev. Mod. Phys., 55, 403-447 (1983).6. Tajima, T., Computational Plasma Physics: With Applications to Fusion and Astrophysics, Addison-

Wesley Publishing Company, Inc., Redwood City, 1989.7. Villasenor, J., and Buneman, O., Computer Physics Communications, 69, 306-316 (1992).8. Koga, J., Nakajima, K., Yamagiwa, M., and Zhidkov, A., these proceedings (2002).9. Yabe, T., and Aoki, T., Computer Physics Communications, 66, 219-232 (1991).10. Yabe, T., Ishikawa, T., and Wang, P. Y, Computer Physics Communications, 66, 233-242 (1991).11. Berz, M., Particle Accelerators, 24, 109^124 (1989).12. Utsumi, T., Kunugi, T., and Koga, J., Computer Physics Communications, 108, 159-179 (1998).

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