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Recently it has been suggested that the fragmentation boundary in SmoothedParticle Hydrodynamic (SPH) and FARGO simulations of self-gravitating accretiondiscs with beta-cooling do not converge as resolution is increased.Furthermore, this recent work suggests that by carefully optimising theartificial viscosity parameters in these codes it can be shown thatfragmentation may occur for much longer cooling times than earlier worksuggests. If correct, this result is intriguing as it suggests that gas giantplanets could form, via direct gravitational collapse, reasonably close totheir parent stars. This result is, however, slightly surprising and there havebeen a number of recent studies suggesting that the result is likely anindication of a numerical problem with the simulations. One suggestion, inparticular, is that the SPH results are influenced by the manner in which thecooling is implemented. We extend this work here and show that if the coolingis implemented in a manner that removes a known numerical artefact in the shockregions, the fragmentation boundary converges to a value consistent withearlier work and that fragmentation is unlikely for the long cooling timessuggested by this recent work. We also investigate the optimisation of theartificial viscosity parameters and show that the values that appear optimalare likely introducing numerical problems in both the SPH and FARGOsimulations. We therefore conclude that earlier predictions for the coolingtimes required for fragmentation are likely correct and that, as suggested bythis earlier work, fragmentation cannot occur in the inner parts (r < 50 au) oftypical protostellar discs.

Convergence of simulations of self-gravitating accretion discs II: Sensitivity to the implementation of radiative cooling and artificial viscosity