A Particle Method without Remeshing

TL;DR

This paper introduces a simple modification to particle methods that eliminates the need for remeshing, achieves optimal error bounds, and demonstrates high accuracy and stability in fluid dynamics simulations.

Contribution

A novel tweak to particle regularisation schemes enabling optimal error bounds without remeshing, validated on advection and Navier--Stokes equations.

Findings

Achieves optimal error bounds of order ^n without remeshing.

Particle methods show high accuracy and long-term stability.

Competitive performance with discontinuous Galerkin methods.

Abstract

We propose a simple tweak to a recently developed regularisation scheme for particle methods. This allows us to chose the particle spacing $h$ proportional to the regularisation length $\sigma$ and achieve optimal error bounds of the form $\mathcal{O}(\sigma^n)$, $n\in\mathbb{N}$, without any need of remeshing. We prove this result for the linear advection equation but also carry out high-order experiments on the full Navier--Stokes equations. In our experiments the particle methods proved to be highly accurate, long-term stable, and competitive with discontinuous Galerkin methods.