  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.
