  Recently it has been suggested that the fragmentation boundary in Smoothed
Particle Hydrodynamic (SPH) and FARGO simulations of self-gravitating accretion
discs with beta-cooling do not converge as resolution is increased.
Furthermore, this recent work suggests that by carefully optimising the
artificial viscosity parameters in these codes it can be shown that
fragmentation may occur for much longer cooling times than earlier work
suggests. If correct, this result is intriguing as it suggests that gas giant
planets could form, via direct gravitational collapse, reasonably close to
their parent stars. This result is, however, slightly surprising and there have
been a number of recent studies suggesting that the result is likely an
indication of a numerical problem with the simulations. One suggestion, in
particular, is that the SPH results are influenced by the manner in which the
cooling is implemented. We extend this work here and show that if the cooling
is implemented in a manner that removes a known numerical artefact in the shock
regions, the fragmentation boundary converges to a value consistent with
earlier work and that fragmentation is unlikely for the long cooling times
suggested by this recent work. We also investigate the optimisation of the
artificial viscosity parameters and show that the values that appear optimal
are likely introducing numerical problems in both the SPH and FARGO
simulations. We therefore conclude that earlier predictions for the cooling
times required for fragmentation are likely correct and that, as suggested by
this earlier work, fragmentation cannot occur in the inner parts (r < 50 au) of
typical protostellar discs.

Earth and Planetary Astrophysics;Solar and Stellar Astrophysics