Black holes accreting well below the Eddington rate are believed to have
geometrically thick, optically thin, rotationally supported accretion discs in
which the Coulomb mean free path is large compared to $GM/c^2$. In such an
environment, the disc evolution may differ significantly from ideal
magnetohydrodynamic predictions. We present non-ideal global axisymmetric
simulations of geometrically thick discs around a rotating black hole. The
simulations are carried out using a new code ${\rm\it grim}$, which evolves a
covariant extended magnetohydrodynamics model derived by treating non-ideal
effects as a perturbation of ideal magnetohydrodynamics. Non-ideal effects are
modeled through heat conduction along magnetic field lines, and a difference
between the pressure parallel and perpendicular to the field lines. The model
relies on an effective collisionality in the disc from wave-particle scattering
and velocity-space (mirror and firehose) instabilities. We find that the
pressure anisotropy grows to match the magnetic pressure, at which point it
saturates due to the mirror instability. The pressure anisotropy produces
outward angular momentum transport with a magnitude comparable to that of MHD
turbulence in the disc, and a significant increase in the temperature in the
wall of the jet. We also find that, at least in our axisymmetric simulations,
conduction has a small effect on the disc evolution because (1) the heat flux
is constrained to be parallel to the field and the field is close to
perpendicular to temperature gradients, and (2) the heat flux is choked by an
increase in effective collisionality associated with the mirror instability.
