  Turbulent magnetic diffusivity plays an important role for accretion disks
and the launching of disk winds. We have implemented magnetic diffusivity,
respective resistivity in the general relativistic MHD code HARM. This paper
describes the theoretical background of our implementation, its numerical
realization, our numerical tests and preliminary applications. The test
simulations of the new code rHARM are compared with an analytic solution of the
diffusion equation and a classical shock tube problem. We have further
investigated the evolution of the magneto-rotational instability (MRI) in tori
around black holes for a range of magnetic diffusivities. We find indication
for a critical magnetic diffusivity (for our setup) beyond which no MRI
develops in the linear regime and for which accretion of torus material to the
black hole is delayed. Preliminary simulations of magnetically diffusive thin
accretion disks around Schwarzschild black holes that are threaded by a
large-scale poloidal magnetic field show the launching of disk winds with mass
fluxes of about 50% of the accretion rate. The disk magnetic diffusivity allows
for efficient disk accretion that replenishes the mass reservoir of the inner
disk area and thus allows for long-term simulations of wind launching for more
than 5000 time units.
