  Abstract: Assuming that the numerical diffusivity triggered by violations of
the force-free electrodynamics constraints is a proxy for the physical
resistivity, we examine its impact on the overall dynamics of force-free
aligned pulsar magnetospheres endowed with an equatorial current sheet. We
assess the constraint violations as a diffusivity source. The effects of
modifications on electric fields used to restore force-free conditions are not
confined to the equatorial current sheet, but modify the magnetospheric
dynamics on timescales shorter than the pulsar rotational period. These
corrections propagate especially via a channel that was unexplored, namely,
changes induced to the electric charge density, $\rho$. We quantify the global
consequences of diffusivity by comparing different techniques to model $\rho$.
By default, we combine a conservative $\rho$-evolution with
hyperbolic/parabolic cleaning of inaccuracies in the Maxwell equations. As an
alternative, we enforce a constrained evolution where $\rho$ is directly
computed as the electric field divergence. The conservative approach reduces
the Poynting flux dissipated in the equatorial current sheet by an order of
magnitude, along with an increase of the pulsar luminosity driven by a shift of
the Y-point location. The luminosity changes according to $L_{\rm Y}\propto
\alpha^{0.11}$, where $\alpha$ is the ratio of diffusion to advection
timescales, controlling the amount of (numerical) diffusivity. Our models
suggest interpreting the luminosity dependence on the Y-point location as
differences in resistivities encountered at the equatorial current sheet.
Alternatively, they could be interpreted in terms of the pair formation
multiplicity, $\kappa$, smaller diffusion being consistent with $\kappa\gg 1$.
