  We numerically construct slim, global, vertically integrated models of
optically thin, transonic accretion discs around black holes, assuming a
regularity condition at the sonic radius and boundary conditions at the outer
radius of the disc and near the black hole. In agreement with several previous
studies, we find two branches of shock-free solutions, in which the cooling is
dominated either by advection, or by local radiation. We also confirm that the
part of the accretion flow where advection dominates is in some circumstances
limited in size: it does not extend beyond a certain outer limiting radius. New
results found in our paper concern the location of the limiting radius and
properties of the flow near to it. In particular, we find that beyond the
limiting radius, the advective dominated solutions match on to Shapiro,
Lightman & Eardley (SLE) discs through a smooth transition region. Therefore,
the full global solutions are shock-free and unlimited in size. There is no
need for postulating an extra physical effect (e.g. evaporation) for triggering
the ADAF-SLE transition. It occurs due to standard accretion processes
described by the classic slim disc equations.
