  Nonlinear time-dependent calculations have been carried out in order to study
the evolution of the thermal instability for optically thick, transonic, slim
accretion discs around black holes. In the present calculations we have
investigated only the original version of the slim disc model with low
viscosity. This version does not yet contain several important non-local
effects but our aim is to use it as a standard reference against which to
compare the results from forthcoming studies in which additional effects will
be added one by one thus giving a systematic way of understanding the
contribution from each of them. A range of results for different cases is
presented showing a number of interesting features. One preliminary conclusion
is that the stabilizing effect of advection seems not to be strong enough in
these low viscosity models to allow for limit cycle behaviour to occur.
