Numerical Convergence in Smoothed Particle Hydrodynamics
  We study the convergence properties of smoothed particle hydrodynamics (SPH)
using numerical tests and simple analytic considerations. Our analysis shows
that formal numerical convergence is possible in SPH only in the joint limit $N
\rightarrow \infty$, $h \rightarrow 0$, and $N_{nb} \rightarrow \infty$, where
$N$ is the total number of particles, $h$ is the smoothing length, and $N_{nb}$
is the number of neighbor particles within the smoothing volume used to compute
smoothed estimates. Previous work has generally assumed that the conditions $N
\rightarrow \infty$ and $h \rightarrow 0$ are sufficient to achieve
convergence, while holding $N_{nb}$ fixed. We demonstrate that if $N_{nb}$ is
held fixed as the resolution is increased, there will be a residual source of
error that does not vanish as $N \rightarrow \infty$ and $h \rightarrow 0$.
Formal numerical convergence in SPH is possible only if $N_{nb}$ is increased
systematically as the resolution is improved. Using analytic arguments, we
derive an optimal compromise scaling for $N_{nb}$ by requiring that this source
of error balance that present in the smoothing procedure. For typical choices
of the smoothing kernel, we find $N_{nb} \propto N^{1/2}$. This means that if
SPH is to be used as a numerically convergent method, the required
computational cost does not scale with particle number as $O(N)$, but rather as
$O(N^{1+\delta})$, where $\delta \approx 1/2$, with a weak dependence on the
form of the smoothing kernel.
