  We review recent results of SPH simulations of gravitational instability in
gaseous protoplanetary disks,emphasizing the role of thermodynamics in both
isolated and binary systems. Contradictory results appeared in the literature
regarding disk fragmentation at tens of AU from the central star are likely due
to the different treatment of radiation physics as well as reflecting different
initial conditions. Further progress on the subject requires extensive
comparisons between different codes with the requirement that the same initial
conditions are adopted. It is discussed how the local conditions of the disks
undergoing fragmentation at $R < 25$ AU in recent SPH simulations are in rough
agreement with the prediction of analytical models, with small differences
being likely related to the inability of analytical models to account for the
dynamics and thermodynamics of three-dimensional spiral shocks. We report that
radically different adaptive hydrodynamical codes, SPH and adaptive mesh
refinement (AMR), yield very similar results on disk fragmentation at
comparable resolution in the simple case of an isothermal equation of state. A
high number of refinements in AMR codes is necessary but not sufficient to
correctly follow fragmentation, rather an initial resolution of the grid high
enough to capture the wavelength of the strongest spiral modes when they are
still barely nonlinear is essential. These tests represent a useful benchmark
and a starting point for a forthcoming code comparison with realistic radiation
physics.
