A new convergence analysis of the particle method for the Camassa-Holm equation

TL;DR

This paper introduces a new, easier-to-apply convergence analysis for the particle method used in solving PDEs like the Camassa-Holm equation, expanding on previous compactness-based approaches.

Contribution

It develops a self-contained convergence proof using measure-valued functions and the bounded Lipschitz metric, simplifying previous methods.

Findings

Convergence and regularity results are recovered as a consequence.

The new analysis is computationally easier to establish.

Applicable to a range of PDEs including the Camassa-Holm equation.

Abstract

We present a new self-contained convergence analysis of the particle method that can be applied to a range of PDEs, including the Camassa-Holm equation. It is a development of the analysis of Chertock, Liu and Pendleton, which used compactness properties of spaces of functions having bounded variation. In our analysis we establish solutions by applying a metric Arzel\`a-Ascoli compactness result to a space of measure-valued functions equipped with the bounded Lipschitz metric. All the convergence and regularity results of the previous analysis follow as a consequence and are computationally easier to establish.