I am trying to write a PIC (Particle In Cell) code to simulate plasma physics. I am starting with the simplest case, which is a 1D system with a longitudinal field $E = E(x)$. I am using this article as reference (it has some typos, though)
- Hui-Chun Wu, JPIC & How to make a PIC code, arXiv:1104.3163.
In it, finite differences are applied to Gauss Law, which gives:
$$E(x + dx, t) =dx \rho(x,t)/\epsilon_0 + E(x,t)$$
(The article uses another normalisation) I would expect the electric field to vanish as x increases. However, this equation seems to say that if $\rho > 0$, then $E(x + dx) > E(x), \forall x$. This would be the result even for the simplest case, i.e., a single point-charge. Where is the mistake? Thank you!