  Accretion discs around black holes can become warped by Lense-Thirring
precession if the disc is tilted with respect to the black hole spin vector.
When the disc viscosity is sufficiently large that warp propagation is
diffusive, the inner disc can align with the black hole spin. However, if the
viscosity is small, such that the warp propagates as a wave, then steady-state
solutions to the linearised fluid equations exhibit an oscillatory radial
profile of the disc tilt angle close to the black hole. Here we show, for the
first time, that these solutions are in good agreement with three-dimensional
hydrodynamical simulations, in which the viscosity is isotropic and measured to
be small compared to the disc angular semi-thickness, and in the case that the
disc tilt -- and thus the warp amplitude -- remains small. We show using both
the linearised fluid equations and hydrodynamical simulations that the inner
disc tilt can be more than several times larger than the original disc tilt,
and we provide physical reasoning for this effect. We explore the transition in
disc behaviour as the misalignment angle is increased, finding increased
dissipation associated with regions of strong warping. For large enough
misalignments the disc becomes unstable to disc tearing and breaks into
discrete planes. For the simulations we present here, we show that the total
(physical and numerical) viscosity at the time the disc breaks is small enough
that the disc tearing occurs in the wave-like regime, substantiating that disc
tearing is possible in this region of parameter space. Our simulations
demonstrate that high spatial resolution, and thus low numerical viscosity, is
required to accurately model the warp dynamics in this regime. Finally, we
discuss the observational implications of our results.
