From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

TL;DR

This paper systematically derives equations of motion for regularized continuous media from the principle of least action, compares different regularization orders, and proves convergence of SPH methods with numerical validation.

Contribution

It provides a rigorous derivation of SPH equations from variational principles and establishes convergence results in the Wasserstein distance for a broad class of force fields.

Findings

Derived two different equations of motion depending on regularization order.

Proved convergence of measure-valued solutions in Wasserstein distance.

Numerically validated convergence with 1D and 2D examples.

Abstract

In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method (e.g. Monaghan), and with the equation treated by Di Lisio et al., respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.