pose serious numerical limitations on the time evolution of the equations. For this reason, a distinct class of implicit-explicit evolution methods has been developed, the RKIMEX schemes [@pareschi_2005_ier], which we have presented in detail in Ref. [@Dionysopoulou:2012pp].

### Generalized Ohm’s law

To close the system of equations presented above, a relation for the electric current density in terms of the other fields is necessary, just like Ohm’s law provides a prescription for the spatial conduction current to be proportional to the electric field. A generalized Ohm’s law provides the necessary coupling of the electric current density to the electromagnetic and matter fields. Previous work toward relativistic versions of the generalized Ohm’s law includes the investigations in Refs. [@Blackman1994; @Khanna1998; @Kandus2008; @Andersson2012]. In one of the simpler cases the spatial conduction current can be considered as proportional to the electric field measured by the comoving observer. Therefore the electric four-current density can be written as the superposition of an advective and a conductive current [@Tsamparlisbook], which takes the form of the generalized Ohm’s law $$I^{\mu} = \tilde{J}^{\mu}_{\rm adv} + \tilde{J}^{\mu}_{\rm cond} = 
\tilde{q} u^\mu + \tilde{\sigma}^{\mu\nu}e_\nu\,.
\label{eq:localcurrent}$$ Here, $\tilde{q}:= -I^{\mu} u_{\mu} = [q + (\tilde{J}_a n^a)]/W$ is the electric charge density 