  We study the consistency and convergence of smoothed particle hydrodynamics
(SPH), as a function of the interpolation parameters, namely the number of
particles $N$, the number of neighbors $n$, and the smoothing length $h$, using
simulations of the collapse and fragmentation of protostellar rotating cores.
The calculations are made using a modified version of the GADGET-2 code that
employs an improved scheme for the artificial viscosity and power-law
dependences of $n$ and $h$ on $N$, as was recently proposed by Zhu et al.,
which comply with the combined limit $N\to\infty$, $h\to 0$, and $n\to\infty$
with $n/N\to 0$ for full SPH consistency, as the domain resolution is
increased. We apply this realization to the "standard isothermal test case" in
the variant calculated by Burkert & Bodenheimer and the Gaussian cloud model of
Boss to investigate the response of the method to adaptive smoothing lengths in
the presence of large density and pressure gradients. The degree of consistency
is measured by tracking how well the estimates of the consistency integral
relations reproduce their continuous counterparts. In particular, $C^{0}$ and
$C^{1}$ particle consistency is demonstrated, meaning that the calculations are
close to second-order accuracy. As long as $n$ is increased with $N$, mass
resolution also improves as the minimum resolvable mass $M_{\rm min}\sim
n^{-1}$. This aspect allows proper calculation of small-scale structures in the
flow associated with the formation and instability of protostellar disks around
the growing fragments, which are seen to develop a spiral structure and
fragment into close binary/multiple systems as supported by recent
observations.
