Numerical Convergence in Smoothed Particle Hydrodynamics

TL;DR

This paper investigates the convergence properties of smoothed particle hydrodynamics (SPH), showing that true convergence requires increasing the number of neighbor particles with resolution, which impacts computational cost.

Contribution

It demonstrates that SPH convergence depends on scaling the neighbor count with particle number, providing an optimal scaling law and analyzing its computational implications.

Findings

Formal convergence requires increasing neighbor particles with resolution.

Optimal neighbor scaling is proportional to the square root of total particles.

Computational cost scales as N^{1.5} for convergent SPH simulations.

Abstract

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.