Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on an implicit in time stepping and a finite element or finite difference discretization in space. Inherent features include positivity, energy decrease and mesh adaptation in the case of blow up densities or compactly supported solutions.