  We carry out simulations of gravitationally unstable discs using a Smoothed
Particle Hydrodynamics (SPH) code and a grid-based hydrodynamics code, FARGO,
to understand the previous non-convergent results reported by Meru & Bate
(2011a). We obtain evidence that convergence with increasing resolution occurs
with both SPH and FARGO and in both cases we find that the critical cooling
timescale is larger than previously thought. We show that SPH has a first-order
convergence rate while FARGO converges with a second-order rate. We show that
the convergence of the critical cooling timescale for fragmentation depends
largely on the numerical viscosity employed in both SPH and FARGO. With SPH,
particle velocity dispersion may also play a role. We show that reducing the
dissipation from the numerical viscosity leads to larger values of the critical
cooling time at a given resolution. For SPH, we find that the effect of the
dissipation due to the numerical viscosity is somewhat larger than had
previously been appreciated. In particular, we show that using a quadratic term
in the SPH artificial viscosity (beta_{SPH}) that is too low appears to lead to
excess dissipation in gravitationally unstable discs, which may affect any
results that sensitively depend on the thermodynamics, such as disc
fragmentation. We show that the two codes converge to values of the critical
cooling timescale, beta_{crit} > 20 (for a ratio of specific heats of
gamma=5/3), and perhaps even as large as beta_{crit} \approx 30. These are
approximately 3-5 times larger than has been found by most previous studies.
This is equivalent to a maximum gravitational stress that a disc can withstand
without fragmenting of alpha_{GI,crit} \approx 0.013-0.02, which is much
smaller than the values typically used in the literature. It is therefore
easier for self-gravitating discs to fragment than has been concluded from most
past studies.
