Mathematics of Smoothed Particle Hydrodynamics: a Study via Nonlocal Stokes Equations

TL;DR

This paper analyzes a nonlocal continuum model for the Stokes equations used in Smoothed Particle Hydrodynamics, establishing well-posedness and the connection to local models as the nonlocal interaction range diminishes.

Contribution

It introduces a nonlocal relaxation of the Stokes system, proving its well-posedness and convergence to the classical model as the nonlocality parameter tends to zero.

Findings

Nonlocal Stokes model is well-posed for certain operators.

The nonlocal model converges to the classical Stokes equations as interaction range shrinks.

Implications for designing numerical methods based on nonlocal models.

Abstract

Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length $\delta$ which measures the range of nonlocal interactions. It serves as a bridge between the discrete approximation schemes that involve a nonlocal integral relaxation and the local continuum models. We show that for a class of carefully chosen nonlocal operators, the resulting nonlocal Stokes equation is well-posed and recovers the original Stokes equation in the local limit when $\delta$ approaches zero. We also discuss the implications of our finding on the design of numerical methods.