Eulerian frame coming from the generalized Ohm’s law can be simplified so that $$\begin{aligned}
\label{eq:ohm_law}
J^i = q v^i + W \sigma [E^i + \epsilon^{ijk} v_j B_k - (v_k E^k) v^i]\,,\end{aligned}$$ where we used the fact that the four-current density can be also written as $I^{\mu}=q n^\mu + J^{\mu}$. In our simulations, the conductivity $\sigma$ is chosen to be either a constant or a function of the rest-mass density and a discussion will be presented in Sec. \[sec:rmhdatmo\]. The last term in Eq.  represents the Hall effects, which, however, we set to zero for simplicity.

Numerical Setup {#sec:numsetup}
===============

Field equations {#sec:nr:fe}
---------------

The evolution of the spacetime ([i.e., ]{}of the 3-metric, extrinsic curvature, and conformal factor) is obtained using the `McLachlan` code, which implements the BSSNOK formulation of the Einstein equations [@Nakamura87; @Shibata95; @Baumgarte99] employing three-dimensional finite-differencing operators for calculating the fluxes on the right-hand side of the Einstein equations [@Pollney:2007ss]. The time integration is carried out using the third-order accurate strong stability preserving implicit-explicit Runge–Kutta scheme outlined in Ref. [@Dionysopoulou:2012pp]. The time step on each grid is limited by the Courant–Friedrichs–Lewy (CFL) condition [@Rezzolla_book:2013] and hence the Courant coefficient is set to be $0.25$ on all refinement levels. 