On the kernel and particle consistency in smoothed particle hydrodynamics

TL;DR

This paper investigates the consistency of smoothed particle hydrodynamics (SPH), demonstrating it has a second-order convergence rate and that proper limiting procedures restore full consistency and accuracy.

Contribution

The study clarifies the convergence order of SPH and shows that joint limits of particle number, smoothing length, and neighbors restore consistency and accuracy.

Findings

SPH has a limiting second-order convergence rate.

Corrective SPH schemes are at best first or second order accurate.

Joint limits restore full particle consistency.

Abstract

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.