  This paper presents our latest numerical simulations of accretion disks that
are misaligned with respect to the rotation axis of a Kerr black hole. In this
work we use a new, fully conservative version of the Cosmos++ general
relativistic magnetohydrodynamics (GRMHD) code, coupled with an ad hoc cooling
function designed to control the thickness of the disk. Together these allow us
to simulate the thinnest tilted accretion disks ever using a GRMHD code. In
this way, we are able to probe the regime where the dimensionless stress and
scale height of the disk become comparable. We present results for both
prograde and retrograde cases. The simulated prograde tilted disk shows no sign
of Bardeen-Petterson alignment even in the innermost parts of the disk. The
simulated retrograde tilted disk, however, does show modest alignment. The
implication of these results is that the parameter space associated with
Bardeen-Petterson alignment for prograde disks may be rather small, only
including very thin disks. Unlike our previous work, we find no evidence for
standing shocks in our simulated tilted disks. We ascribe this to the
combination of small black hole spin, small tilt angle, and small disk scale
height in these simulations. We also add to the growing body of literature
pointing out that the turbulence driven by the magnetorotational instability in
global simulations of accretion disks is not isotropic. Finally, we provide a
comparison between our moderately thin, untilted reference simulation and other
numerical simulations of thin disks in the literature.
