Finite volume methods on spheres and spherical centroidal Voronoi meshes

Abstract

We study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular, the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the L2 error of the approximate solution is of quadratic order when the underlying Voronoi mesh is given by a spherical centroidal Voronoi mesh. We also demonstrates numerically the high accuracy and the superconvergence of the approximate solutions.