A new insight into the consistency of smoothed particle hydrodynamics

TL;DR

This paper provides a rigorous analysis of the consistency of smoothed particle hydrodynamics (SPH), deriving new integral relations and error bounds that clarify the conditions for convergence to the continuum limit.

Contribution

It introduces a novel error analysis framework for SPH using the Poisson summation formula, establishing new consistency relations and convergence conditions.

Findings

Particle approximation converges to kernel approximation as the number of particles increases.

Consistency conditions depend explicitly on smoothing length and particle count.

A dominant error term decreases as the number of particles grows, confirming long-standing conjectures.

Abstract

In this paper the problem of consistency of smoothed particle hydrodynamics (SPH) is solved. A novel error analysis is developed in $n$-dimensional space using the Poisson summation formula, which enables the treatment of the kernel and particle approximation errors in combined fashion. New consistency integral relations are derived for the particle approximation which correspond to the cosine Fourier transform of the classically known consistency conditions for the kernel approximation. The functional dependence of the error bounds on the SPH interpolation parameters, namely the smoothing length $h$ and the number of particles within the kernel support ${\cal{N}}$ is demonstrated explicitly from which consistency conditions are seen to follow naturally. As ${\cal{N}}\to\infty$, the particle approximation converges to the kernel approximation independently of $h$ provided that the particle mass scales with $h$ as $m\propto h^{\beta}$, with $\beta >n$. This implies that as $h\to 0$, the joint limit $m\to 0$, ${\cal{N}}\to\infty$, and $N\to\infty$ is necessary for complete convergence to the continuum, where $N$ is the total number of particles. The analysis also reveals the presence of a dominant error term of the form $(\ln {\cal{N}})^{n}/{\cal{N}}$, which tends asymptotically to $1/{\cal{N}}$ when ${\cal{N}}\gg 1$, as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates.