Review and Comparison of Particle-in-Cell and
Vlasov Simulation methods with application to
relativistic self-focusing
James 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 methods
for plasma simulation with applications to relativistic self-focusing of high intensity laser pulses in
plasmas.
INTRODUCTION
Plasma phenomena occur throughout nature and are the result of the complex nature
of the collective interaction of many charged particles. Simulation of plasmas on large
scale computers has become an invaluable tool in analyzing various aspects of plasma
behavior. In particular for laser plasma interaction it has become the dominant means
of explaining results of experiments using high intensity short pulse lasers. In this paper
we will be discussing two types of plasma simulation techniques. They are particle and
Vlasov simulation techniques.
PARTICLE SIMULATION
In the real world individual charged particles in a plasma are coupled to each other via
electromagnetic fields (E,B). Particles are accelerated by the electromagnetic fields via
the Lorentz force equation:
dt~jm
(2)
()
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 simulation
particles are advanced by these equations.
Typically a finite differencing scheme is used to advance the particles which was
developed by Boris [1], First, we rewrite equation 1 in the form:
dt
m
jc
CP634, Science of Super strong Field Inter actions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
388
where u = yv. Finite differencing this equation we get:
-un~2
Af
q ,^T?nn j_
— ^
2"+__2 +un~
(4)
where the superscript n refers to whole time steps and n±^ refers to fractional time
steps. This equation contains both E and B. One can eliminate E by expressing introducing the following variables:
(5)
2m
where y1 = 1 + (^-) . Rewriting equation 3 we get:
(6)
where
4
M+ =
I
(7)
«+
«z+
«7
Equation 6 represents a matrix equation which can be inverted to get:
1
„ _
(8)
, (9)
Q2 where Q2 - Q2 + Q2 + Q2, Q^ = ^, Qy - ^, and Dz
. Using equations 5
and 8 we can get w n + 2. Finite differencing equation 2 we get:
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
I _
ret
3x{(n-p)xp}
(J,r) =\nxE]
L
C
389
J re/
(11)
where E(x,t) and B(x,t) are the electric and magnetic fields, respectively, generated by
charged particles other than the particle which feels the field. Here, ret refers to the time
in the past where the trajectory of the other particle intersects with the light cone of the
particle which sees the fields, n is the unit normal vector between the particle and other
particle's past position, (3 and y are the usual relativistic factors, and p is d$/dt which is
the usual acceleration divided by c.
From a computational point of view it can be seen that if there are N particles which
interact via the Lienard-Wiechert fields then N2 interactions must be calculated. Resultingly, the amount of computation increases rapidly with particle number so only a
limited number of particles can be calculated in a reasonable time even using supercomputers.
One way of getting around the amount of computation required from direct particleparticle interaction simulations is to compute the electromagnetic fields on a finite number of grids. This method is called Particle-in-Cell (PIC). Many excellent references can
be found describing this method [3, 4, 5, 6] so in this section we will only cover briefly
the 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 equations are solved:
V - £ = 47tp
VxE = --c^
V-B = Q
(12)
Vx£=f/+If
(13)
where p refers to the charge density and /refers to the current density. The charge and
current density are accumulated on the grid from the particles. By using grids instead of
calculating direct interactions the number of calculations for a N particle system goes
as [6]: MlnM + bN where M is the number of grids and b is a constant. The increase in
computation 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 include
Fast Fourier Transforms [5] and implicit finite difference schemes [4]. We will describe
in more detail an explicit finite difference scheme [3] which is more suitable for implementation on massively parallel computers where local solutions are optimal for speed.
Rearranging Maxwell's equations where equations 12 are taken as initial conditions and
finite differencing each component of the electric field (Ex, Ey,Ez) for a two dimensional
grid we get:
A/
A;y
n
p +5
D
Z. • , 1 , , 1
_ p
D
7-
2
1
(15)
390
HI , 1
k+
2
Ay
J, £? B,
5,
B, J,
J s
, ,
*,
k •
k
I
2
*~t^
Bs
5
r^J
T
Ev Bt
*. ^i B,
^
J
B,
*i ^.
^r
J* *,
5X i
E* R
f
4\
5
r
5r
^ ^,
\
5,
Y
,r
R
N'
j
J
*. J.
V 1 -
;
FIGURE 1. The finite difference positions of the fields on a uniform two dimensional grid is shown.
• H+l
=
c-
B*
— c-
Ax
(16)
and for the magnetic field (BX:By,Bz) we get
Ar
=
— c77 n ]
Z i_u fr-u !
Z5v
J7z n
i ! t_u l
7-2^+2
7+2'^+2
Ar
(18)
Ax
n
A?
(17)
A};
=
T? n
-c(-
Ax
A^
In figure 1 we show the sequence of calculation for the fields on a uniform two
dimensional grid. Note that the E and B fields are offset from one another by half time
steps and half a grid cell. This finite differencing scheme is stable as long as the Courant
condition is satisfied for the simulation time step Ar. In the case of two dimensions the
condition is[3]: cAf < Ax/\/2 assuming Ax = Ay where Ax and A;y are the grid sizes in
the x and y direction, respectively.
The current terms (Jx,Jy,Jz) in equations 14, 15, and 16 are calculated by accumulating the current contributions from the simulation particles onto the grid. By appropriately accumulating current on the grid one can maintain charge conservation without having to recalculate equation 12. The technique is fairly detailed so we refer the
reader to the reference [7]. Figure 2 shows the collection of current in the simplest
391
(x+6x9y+6y)
FIGURE 2. The collection of current from a particle onto a uniform two dimensional grid (left) and the
interpolation of the fields on the grid to the particle (right) are shown. (x,y) is the initial particle position
and (jc + 5jc,y + 5y) is the final position
case where four cell boundaries are crossed by the particle. The currents are calcuand
lated as: Jxl =
Jyi =
JX2 =
Jy2 =
+ Jt-h 5&c), where 8jc and &y refer to the change in the particle position in
one time step in the x and ;y directions, respectively. There are more complicated crossings of 7 and 10 boundaries which are described in [7].
Once the new fields have been calculated on the uniform grid, they need to be
interpolated 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,fc + l)5^8^c where / represents the field quantity being
interpolated to the particle position.
In addition to the various proceedures described above additional constraints are
placed on the simulation due to numerical instabilities. One instability is the thermal
instability. If the temperature of the plasma particles is not high enough then the plasma
will numerically heat up unless the following condition is met [3]: ^ > 0.3 where
X/) = ^/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 proceedings dealing with the study of proton acceleration and relativistic self-focusing by this
author [8].
VLASOV SIMULATION
In this section we describe Vlasov simulation methods. In the previous section we talked
about combinations of simulation particles and grids to model plasma behavior. In the
case of Vlasov simulation only grids are used to model the plasma. The advantage of
392
V
max
- ———— ——
1————————————
V
AV
»< Y
0
L
Ax
V
-
FIGURE 3. The grid on which the Vlasov equations are solved. The grid has 2M + 1 cells in velocity V
with 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 only
with grids parallelization on massively parallel computers is fairly straightforward. The
disadvantage of this technique is that large amounts of computer memory are needed
and different types of numerical instabilities occur. In this section we describe numerical
solution 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, t
is the time, and E is the electrostatic field E(x,t). The following normalization is
used:Ax —)• A,£>, Ar —> co^ = ^/4nnoe2/m, v —> A^co^. Equations 20 and 21 are solved
on a uniform grid which is shown in figure 3.
Equation 20 is a hyperbolic equation so we can use the cubic interpolation spline
technique (CIP)[9,10]. In addition to increase accuracy we use differential algebra (DA)
which allows one to calculate derivatives algebraically, see [11]. This combination is
called the DA-CIP scheme[12]. In the following section we will briefly describe this
method. 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)
393
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 equation:
(28)
where A? is the time step size. We can calculate this equation via a second order RungeKutta 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 this
we need to calculate the electric field E. We know that: E — — -^ where (|) is the scalar
potential so that after finite differencing we get £/ = ^JAx/"1 - In addition, we can get
dxEi = ~
~A^2+^~ where the indicies / are the same as figure 3. Equation 21 can
then be written in the form:
y*'-' = T /(*, v,r)*- 1 ^
AX,-
•_
•_
J — oo
[F^O*- 1
••
7—
——7
7max ~~ 1
394
(31)
JV;
(32)
m
FIGURE 4. Each cell moves when q(t] is advanced.
This equation represents a tridiagonal matrix which can quickly be solved by the Thomas
algorithm. Once the potential <|) has been solved the electric field can be calculated. In
equation 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 advanced 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 advanced in time each cell position also moves as seen in figure 4. We need to reconstruct
the distribution function from these new cell positions. This can be done by using the
cubic interpolation function in_equation 33_. Replacing the old values, fij^xfij^vfij
with 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
Fij(Xi,Vj+i) =
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)
395
By using this function we can determine the values of the grid points within each new
cell ABCD in figure 4. Let Rnm — (Xn, Vm) represent the grid point in ABCD. In order to
calculate the value at this point we use the cubic function. However, it is a function of
the old cell positions in ABCD. This can be resolved by finding the mapping between
the new cell and old cell. To find the position R° = (X°, V°) in ABCD corresponding to
Rnm 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 linear
transformation matrix defined by equation 39. Once we know this transformation we
can 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 be
determined 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 steps
is reached.
COMPARISON AND CONCLUSION
Figure 5 shows results for PIC (left) and Vlasov (right) simulations with initial conditions at the top and final results at the bottom. The simulation run is for the two stream
instability where initially oppositly flowing electron beams are unstable and merge to
form a vortex in x-v phase space. The parameters of each simulation are somewhat close
to eachother. It can be seen that there are fluctuations in the distribution function for
the PIC simulation whereas in the Vlasov simulation there is none. Each simulation
converges to a single vortex. In the case of the Vlasov simulation the distribution is unchanging after some time. However, the PIC simulation is still evolving in time. It will
be of further study to determine which type of simulation is closer to reality over long
time 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 the
techniques described in the section on Vlasov simulation. I would like to thank Kazuhisa
Nakajima for inviting me to give a review talk concerning plasma simulation and the
students who attended my talk asking many thought provoking questions.
396
FIGURE 5. Comparison of PIC results (left) and the Vlasov results (right) are shown for the two-stream
instability where the initial condtions are at the top and the final states are at the bottom.
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