A Lloyd-stabilized Voronoi particle method

Abstract

We study the stability and consistency of Lloyd's algorithm used in combination with the Lagrangian transport of a density on a Voronoï tessellation. We show that relaxation rates as strong as $O(h^{-1/2})$, where $h$ is the mesh size, give convergence to the continuity equation in Wasserstein distance at the rate $O(h^{1/4})$. Thus the mesh can be kept regular by the correction alone, without the remeshing that Lagrangian methods usually require. An application is proposed for the compressible Euler equations.