A Kreiss–Oliger type dissipation is added to the spacetime evolution equations to ensure that any high-frequency noise produced during the evolution (mostly at the refinement levels boundaries) is damped. The damping parameter in the evolution equation for the shift was carefully chosen to be $0.71$ so that the additional term does not become stiff on the coarser grids.

GR-RMHD equations {#sec:nr:rmhd}
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The general-relativistic RMHD (GR-RMHD) equations are solved with the `WhiskyRMHD` code [@Dionysopoulou:2012pp], which uses high-resolution shock-capturing schemes [@Toro99; @Rezzolla_book:2013]. In particular, the reconstruction of the conserved variables is achieved via the Piecewise Parabolic Method [@Colella84], while the fluxes are calculated through the approximate Riemann solver introduced by Harten–Lax–van Leer–Einfeldt [@Harten83] and which requires only knowledge about the maximum characteristic speeds of the system, [i.e., ]{}the speed of light in this case.

The system of RMHD equations is closed by describing the fluid as ideal and with a $\Gamma$-law EOS $$\label{eq:gamma-law}
  p=\rho\epsilon~(\Gamma-1)\,,$$ where $\Gamma=2$ and $K=123.6$. This EOS is clearly an idealization. Much more sophisticated EOSs have been implemented in our numerical infrastructure [@Galeazzi2013; @Takami:2014; @Takami2015] and have been used by other groups in general-relativistic magnetohydrodynamic (GRMHD) simulations [@Neilsen2014]. However, this idealization is probably adequate at this stage as 