Simulations of Overstable Inertial-acoustic Modes in Black-Hole
  Accretion Discs
  We present two-dimensional inviscid hydrodynamic simulations of overstable
inertial-acoustic oscillation modes (p-modes) in black-hole accretion discs.
These global spiral waves are trapped in the inner-most region of the disc, and
are driven overstable by wave absorption at the corotation resonance ($r_c$)
when the gradient of the background disc vortensity (vorticity divided by
surface density) at $r_c$ is positive and the disc inner boundary is
sufficiently reflective. Previous linear calculations have shown that the
growth rates of these modes can be as high as 10% of the rotation frequency at
the disc inner edge. We confirm these linear growth rates and the primary disc
oscillation frequencies in our simulations when the mode amplitude undergoes
exponential growth. We show that the mode growth saturates when the radial
velocity perturbation becomes comparable to the disc sound speed. During the
saturation stage, the primary disc oscillation frequency differs only slightly
(by less than a few percent) from the linear mode frequency. Sharp features in
the fluid velocity profiles at this stage suggest that the saturation results
from nonlinear wave steepening and mode-mode interactions.
