  We study Lense-Thirring precession of inviscid and viscous misaligned
$\alpha-$discs around a black hole using a gravitomagnetic term in the momentum
equation. For weak misalignments, $i \lesssim 10^{\circ}$, the discs behave
like rigid bodies, undergoing the full suite of classical harmonic oscillator
dynamics including, weak and critically damped motion (due to viscosity),
precession (due to Lense-Thirring torque) and nutation (due to apsidal
precession). For strong misalignments, $i \gtrsim 30^{\circ}$, we find
sufficiently thin, $h/r \lesssim 0.05$ discs break, form a gap and the inner
and outer sub-discs evolve quasi independently apart from slow mass transfer.
Assuming the sound speed sets the communication speed of warps in the disc, we
can estimate the breaking radius by requiring that the inner sub-disc precesses
like a rigid body. We explicitly show for the first time using a grid code that
an Einstein potential is needed to reproduce the analytic properties of the
inner disc edge and find disc breaking. At large inclination angles we find
multiple disc breaks, consistent with recent GRMHD simulations of highly
inclined discs. Our results suggest that the inclusion of a gravitomagnetic
term and appropriate pseudo-Newtonian potential captures the important
quantitative features of misaligned discs.
