  Recent simulations of self-gravitating accretion discs, carried out using a
three-dimensional Smoothed Particle Hydrodynamics (SPH) code by Meru and Bate,
have been interpreted as implying that three-dimensional global discs fragment
much more easily than would be expected from a two-dimensional local model.
Subsequently, global and local two-dimensional models have been shown to
display similar fragmentation properties, leaving it unclear whether the
three-dimensional results reflect a physical effect or a numerical problem
associated with the treatment of cooling or artificial viscosity in SPH. Here,
we study how fragmentation of self-gravitating disc flows in SPH depends upon
the implementation of cooling. We run disc simulations that compare a simple
cooling scheme, in which each particle loses energy based upon its internal
energy per unit mass, with a method in which the cooling is derived from a
smoothed internal energy density field. For the simple per particle cooling
scheme, we find a significant increase in the minimum cooling time scale for
fragmentation with increasing resolution, matching previous results. Switching
to smoothed cooling, however, results in lower critical cooling time scales,
and tentative evidence for convergence at the highest spatial resolution
tested. We conclude that precision studies of fragmentation using SPH require
careful consideration of how cooling (and, probably, artificial viscosity) is
implemented, and that the apparent non-convergence of the fragmentation
boundary seen in prior simulations is likely a numerical effect. In real discs,
where cooling is physically smoothed by radiative transfer effects, the
fragmentation boundary is probably displaced from the two-dimensional value by
a factor that is only of the order of unity.
