Evolution of current density and EM field

Related evolution of a matter density under Newtonian gravity.
Given a Lorentzian manifold representing spacetime, an electromagnetic field is a 2-form $F$ satisfying Maxwell equations

where $J$ is a vector field called current density, which represents the source of the field.

How do we calculate the evolution of $F$ and $J^{\mathrm{♭}}$ with respect to time, provided $J^{\mathrm{♭}}{|}_{t=0}$?

1. Solve Maxwell's equations to determine the initial $F$, given the initial $J$ and any boundary conditions. We are assuming $J$ is constant, with value $J{|}_{t=0}$, for a small time interval.
2. Solve the Lorentz force law or the Vlasov equation to update the motion of the charged particles (and hence $J$).
   1. If modeling discrete particles, use the Lorentz force law to evolve the 4-velocity $u^{\mu }$ of each particle
   2. If modeling a charged fluid, solve the fluid equations coupled with the Lorentz force:
   $${\mathrm{\nabla }}_{\nu }{T}^{\mu \nu }=F_{\nu }^{\mu }{J}^{\nu },$$where $T^{\mu \nu }$ is the energy-momentum tensor of the fluid (is this Vlasov equation?).
3. Use the updated $J$ to recalculate $F$ from Maxwell's equations.
4. Iterate this process, ensuring that the continuity equation is always satisfied.