Impetus: Consistent SPH calculations of 3D spherical Bondi accretion
  onto a black hole
  We present three-dimensional calculations of spherically symmetric Bondi
accretion onto a stationary supermassive black hole (SMBH) of mass $10^{8}$
$M_{\odot}$ within a radial range of $0.02-10$ pc, using a modified version of
the smoothed particle hydrodynamics (SPH) \gad \sp code, which ensures
approximate first-order consistency (i.e., second-order accuracy) for the
particle approximation. First-order consistency is restored by allowing the
number of neighbours, $n_{\rm neigh}$, and the smoothing length, $h$, to vary
with the total number of particles, $N$, such that the asymptotic limits
$n_{\rm neigh}\to\infty$ and $h\to 0$ hold as $N\to\infty$. The ability of the
method to reproduce the isothermal ($\gamma =1$) and adiabatic ($\gamma =5/3$)
Bondi accretion is investigated with increased spatial resolution. In
particular, for the isothermal models the numerical radial profiles closely
match the Bondi solution, except near the accretor, where the density and
radial velocity are slightly underestimated. However, as $n_{\rm neigh}$ is
increased and $h$ is decreased, the calculations approach first-order
consistency and the deviations from the Bondi solution decrease. The density
and radial velocity profiles for the adiabatic models are qualitatively similar
to those for the isothermal Bondi accretion. Steady-state Bondi accretion is
reproduced by the highly resolved consistent models with a percent relative
error of $\lesssim 1$\% for $\gamma =1$ and $\sim 9$\% for $\gamma =5/3$, with
the adiabatic accretion taking longer than the isothermal case to reach steady
flow. The performance of the method is assessed by comparing the results with
those obtained using the standard Gadget and the Gizmo codes.
