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

Convergence of the critical cooling rate for protoplanetary disk fragmentation achieved; the key role of numerical dissipation of angular momentum