Vortices have been proposed as the sites of planet formation, where dust collects and grows into planetesimals, the building blocks of planets. However, for very small dust particles that can be treated as a pressure-less fluid, we have recently discovered the "heavy core" instability, driven by the density gradient in the vortex. In order to understand the eventual outcome of this instability, we need to study its non-linear development. Here, we describe our ongoing work to develop highly accurate numerical models of a vortex with a density gradient embedded within a protoplanetary disk.