  We carry out simulations of gravitationally unstable disks using smoothed
particle hydrodynamics(SPH) and the novel Lagrangian meshless finite mass (MFM)
scheme in the GIZMO code (Hopkins 2015). Our aim is to understand the cause of
the non-convergence of the cooling boundary for fragmentation reported in the
literature. We run SPH simulations with two different artificial viscosity
implementations, and compare them with MFM, which does not employ any
artificial viscosity. With MFM we demonstrate convergence of the critical
cooling time scale for fragmentation at \beta_{crit} =3.. Non-convergence
persists in SPH codes, although it is significantly mitigated with schemes
having reduced artificial viscosity such as inviscid SPH (ISPH) (Cullen &
Dehnen 2010). We show how the non-convergence problem is caused by artificial
fragmentation triggered by excessive dissipation of angular momentum in domains
with large velocity derivatives. With increased resolution such domains become
more prominent. Vorticity lags behind density due to numerical viscous
dissipation in these regions, promoting collapse with longer cooling times.
Such effect is shown to be dominant over the competing tendency of artificial
viscosity to diminish with increasing resolution. When the initial conditions
are first relaxed for several orbits, the flow is more regular, with lower
shear and vorticity in non-axisymmetric regions, aiding convergence. Yet MFM is
the only method that converges exactly. Our findings are of general interest as
numerical dissipation via artificial viscosity or advection errors can also
occur in grid-based codes. Indeed for the FARGO code values of \beta_{crit}
significantly higher than our converged estimate have been reported in the
literature. Finally, we discuss implications for giant planet formation via
disk instability.
