  We are carrying out a programme of non-linear time-dependent numerical
calculations to study the evolution of the thermal instability driven by
radiation pressure in transonic accretion discs around black holes. In our
previous studies we first investigated the original version of the slim-disc
model with low viscosity (parameter alpha = 0.001) for a stellar-mass (10 solar
masses) black hole, comparing the behaviour seen with results from local
stability analysis (which were broadly confirmed). In some of the unstable
models, we saw a violently evolving shock-like feature appearing near to the
sonic point. Next, we retained the original model simplifications but
considered a higher value of alpha = 0.1 and demonstrated the existence of
limit-cycle behaviour under suitable circumstances. The present paper describes
more elaborate calculations with a more physical viscosity prescription and
including a vertically integrated treatment of acceleration in the vertical
direction. Limit-cycle behaviour is still found for a model with alpha = 0.1,
giving a strong motivation to look for its presence in observational data.
