<s>
In	O
plasma	O
physics	O
,	O
the	O
particle-in-cell	B-Algorithm
(	O
PIC	O
)	O
method	O
refers	O
to	O
a	O
technique	O
used	O
to	O
solve	O
a	O
certain	O
class	O
of	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
In	O
this	O
method	O
,	O
individual	O
particles	O
(	O
or	O
fluid	O
elements	O
)	O
in	O
a	O
Lagrangian	O
frame	O
are	O
tracked	O
in	O
continuous	O
phase	O
space	O
,	O
whereas	O
moments	O
of	O
the	O
distribution	O
such	O
as	O
densities	O
and	O
currents	O
are	O
computed	O
simultaneously	O
on	O
Eulerian	B-Algorithm
(	O
stationary	O
)	O
mesh	O
points	O
.	O
</s>
<s>
PIC	B-Algorithm
methods	I-Algorithm
were	O
already	O
in	O
use	O
as	O
early	O
as	O
1955	O
,	O
</s>
<s>
even	O
before	O
the	O
first	O
Fortran	B-Application
compilers	O
were	O
available	O
.	O
</s>
<s>
For	O
many	O
types	O
of	O
problems	O
,	O
the	O
classical	O
PIC	B-Algorithm
method	I-Algorithm
invented	O
by	O
Buneman	O
,	O
Dawson	O
,	O
Hockney	O
,	O
Birdsall	O
,	O
Morse	O
and	O
others	O
is	O
relatively	O
intuitive	O
and	O
straightforward	O
to	O
implement	O
.	O
</s>
<s>
Since	O
the	O
early	O
days	O
,	O
it	O
has	O
been	O
recognized	O
that	O
the	O
PIC	B-Algorithm
method	I-Algorithm
is	O
susceptible	O
to	O
error	O
from	O
so-called	O
discrete	O
particle	O
noise	O
.	O
</s>
<s>
This	O
error	O
is	O
statistical	O
in	O
nature	O
,	O
and	O
today	O
it	O
remains	O
less-well	O
understood	O
than	O
for	O
traditional	O
fixed-grid	O
methods	O
,	O
such	O
as	O
Eulerian	B-Algorithm
or	O
semi-Lagrangian	B-Application
schemes	I-Application
.	O
</s>
<s>
These	O
algorithms	O
use	O
tools	O
of	O
discrete	O
manifold	O
,	O
interpolating	O
differential	O
forms	O
,	O
and	O
canonical	O
or	O
non-canonical	O
symplectic	B-Algorithm
integrators	I-Algorithm
to	O
guarantee	O
gauge	O
invariant	O
and	O
conservation	O
of	O
charge	O
,	O
energy-momentum	O
,	O
and	O
more	O
importantly	O
the	O
infinitely	O
dimensional	O
symplectic	O
structure	O
of	O
the	O
particle-field	O
system	O
.	O
</s>
<s>
In	O
PIC	O
simulation	O
the	O
leapfrog	B-Algorithm
method	I-Algorithm
is	O
used	O
,	O
a	O
second-order	O
explicit	O
method	O
.	O
</s>
<s>
For	O
plasma	O
applications	O
,	O
the	O
leapfrog	B-Algorithm
method	I-Algorithm
takes	O
the	O
following	O
form	O
:	O
</s>
<s>
The	O
global	O
bound	O
on	O
energy	O
error	O
typically	O
associated	O
with	O
symplectic	B-Algorithm
algorithms	I-Algorithm
still	O
holds	O
for	O
the	O
Boris	O
algorithm	O
,	O
making	O
it	O
an	O
effective	O
algorithm	O
for	O
the	O
multi-scale	O
dynamics	O
of	O
plasmas	O
.	O
</s>
<s>
Also	O
spectral	B-Algorithm
methods	I-Algorithm
,	O
such	O
as	O
the	O
fast	O
Fourier	O
transform	O
(	O
FFT	O
)	O
,	O
transform	O
the	O
PDEs	O
into	O
an	O
eigenvalue	O
problem	O
,	O
but	O
this	O
time	O
the	O
basis	O
functions	O
are	O
high	O
order	O
and	O
defined	O
globally	O
over	O
the	O
whole	O
domain	O
.	O
</s>
<s>
The	O
name	O
"	O
particle-in-cell	B-Algorithm
"	O
originates	O
in	O
the	O
way	O
that	O
plasma	O
macro-quantities	O
(	O
number	O
density	O
,	O
current	O
density	O
,	O
etc	O
.	O
)	O
</s>
<s>
Above	O
all	O
,	O
the	O
field	O
interpolation	O
scheme	O
should	O
conserve	O
momentum	B-Algorithm
.	O
</s>
<s>
Simulating	O
the	O
interaction	O
for	O
every	O
pair	O
of	O
a	O
big	O
system	O
would	O
be	O
computationally	O
too	O
expensive	O
,	O
so	O
several	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
have	O
been	O
developed	O
instead	O
.	O
</s>
<s>
For	O
an	O
explicit	O
electromagnetic	O
plasma	O
simulation	O
,	O
the	O
time	O
step	O
must	O
also	O
satisfy	O
the	O
CFL	B-Algorithm
condition	I-Algorithm
:	O
</s>
<s>
Within	O
plasma	O
physics	O
,	O
PIC	O
simulation	O
has	O
been	O
used	O
successfully	O
to	O
study	O
laser-plasma	O
interactions	O
,	O
electron	O
acceleration	O
and	O
ion	O
heating	O
in	O
the	O
auroral	O
ionosphere	O
,	O
magnetohydrodynamics	O
,	O
magnetic	O
reconnection	O
,	O
as	O
well	O
as	O
ion-temperature-gradient	O
and	O
other	O
microinstabilities	O
in	O
tokamaks	B-Application
,	O
furthermore	O
vacuum	O
discharges	O
,	O
and	O
dusty	O
plasmas	O
.	O
</s>
<s>
Hybrid	O
models	O
may	O
use	O
the	O
PIC	B-Algorithm
method	I-Algorithm
for	O
the	O
kinetic	O
treatment	O
of	O
some	O
species	O
,	O
while	O
other	O
species	O
(	O
that	O
are	O
Maxwellian	O
)	O
are	O
simulated	O
with	O
a	O
fluid	O
model	O
.	O
</s>
