Given the long time scale over which our simulations are carried out, reflections due to imperfections of the boundary conditions experience several domain crossings before the end of the simulation. We plan to improve on this in the future by the application of maximally dissipative boundary conditions that minimize the effects of reflections.

### Exterior matching and atmosphere handling {#sec:rmhdatmo}

Our goal with the use of a RMHD framework is that of modelling the exterior of the neutron stars as an electrovacuum, where both the conductivity and the charge density are negligibly small, so that electromagnetic fields should obey in these regions the Maxwell equations in vacuum. On the other hand, we want to model the interior of the stars as highly conducting, so that our equations recover the IMHD limit in such regions. There are several different ways to achieve this; see, for example, Refs. [@Lehner2011; @Dionysopoulou:2012pp; @Palenzuela2013; @Paschalidis2013c]. Each of them has, in our view, its advantages and disadvantages. However, because they all try to model the difficult transition region between two regimes that are intrinsically different, they all represent an approximation. This difficulty is not specific to this problem but is a typical feature of physical problems 