   Next: An example 1D PIC Up: Particle-in-cell codes Previous: Evaluation of electron number

## Solution of Poisson's equation

Consider the solution of Poisson's equation: (303)

where . Note that in normalized units. Let and . We can write   (304)   (305)

which automatically satisfies the periodic boundary conditions and . Note that , since . The other are obtained from (306)

for . The Fourier transformed version of Poisson's equation yields (307)

and (308)

for , where . Finally, (309)

for to , which ensures that the remain real. The discretized version of Eq. (297) is (310)

Of course, and are special cases which can be resolved using the periodic boundary conditions. Finally, suppose that the coordinate of the th electron lies between the th and th grid-points: i.e., . We can then use linear interpolation to evaluate the electric field seen by the th electron: (311)   Next: An example 1D PIC Up: Particle-in-cell codes Previous: Evaluation of electron number
Richard Fitzpatrick 2006-03-29