Hi, Ive been doing some reading into 1 dimensional plasma numerical simulations and they keep referring to solving for a "self-consistent" field. If the simulation is in one dimension with periodic boundary conditions, how would I go about solving this electric field? Example: dE/dx = n - ρ(x) where: n = const = 1 ρ(x) is the charge density and I want to solve for E numerically where E is "self consistent" Thanks for your input.
The electric field depends on a distribution of charges - but the distribution of charges depends on the electric field. This creates a chicken-and-egg situation. A "self consistent" field is one which makes the charges distributed so that they generate the field. We can compute them using an iterative procedure. You start with a guess for a charge distribution ρ_{0}, compute the field that distribution gives rise to. That field will push the charges into a new configuration ρ' - so work out that new distribution as if the field were fixed at what you calculated. Now repeat the procedure for ρ_{1}=(1-λ)ρ_{0}+λρ' where 0<λ<1. You have to guess lambda. Keep going until you keep getting the same result to the desired level of accuracy. The exact method will depend on the context.