On the kernel and particle consistency in smoothed particle
  hydrodynamics
  The problem of consistency of smoothed particle hydrodynamics (SPH) has
demanded considerable attention in the past few years due to the ever
increasing number of applications of the method in many areas of science and
engineering. A loss of consistency leads to an inevitable loss of approximation
accuracy. In this paper, we revisit the issue of SPH kernel and particle
consistency and demonstrate that SPH has a limiting second-order convergence
rate. Numerical experiments with suitably chosen test functions validate this
conclusion. In particular, we find that when using the root mean square error
as a model evaluation statistics, well-known corrective SPH schemes, which were
thought to converge to second, or even higher order, are actually first-order
accurate, or at best close to second order. We also find that observing the
joint limit when $N\to\infty$, $h\to 0$, and $n\to\infty$, as was recently
proposed by Zhu et al., where $N$ is the total number of particles, $h$ is the
smoothing length, and $n$ is the number of neighbor particles, standard SPH
restores full $C^{0}$ particle consistency for both the estimates of the
function and its derivatives and becomes insensitive to particle disorder.
