Recent results in the systematic derivation and convergence of SPH

TL;DR

This paper derives SPH from continuum mechanics principles using a measure-based approach, proves its convergence under certain conditions, and supports these findings with numerical experiments showing convergence as particle number increases.

Contribution

It introduces a measure-based derivation of SPH, provides a convergence proof, and discusses extensions and limitations of previous work.

Findings

SPH converges with respect to Wasserstein distance as particles increase

Numerical experiments show convergence even outside proven conditions

Theoretical convergence may hold under weaker assumptions

Abstract

This paper presents the derivation of SPH from principles of continuum mechanics via a measure-based formu- lation. Additionally, it discusses a theoretical convergence result, the extensions achieved from previous works and the current limitations of the proof. In support of the theoretical result, numerical experiments show that SPH converges with respect to the Wasserstein distance as the number of particles grows to infinity. Convergence is still observed for those numerical experiments which are not covered by the hypotheses of the theoretical result. The latter finding suggests that it should be possible to prove the theoretical result under weaker conditions.