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.
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