NUMERICAL SIMULATION OF DIFFUSIVE AND AGGREGATION PHENOMENA IN NONLINEAR CONTINUITY EQUATIONS BY EVOLVING DIFFEOMORPHISMS

We propose a numerical algorithm for solving nonlinear continuity equations written in Lagrangian coordinates. This transformation is intimately related to variational approaches for the well-posedness of gradient flows of energy functionals with respect to the quadratic transportation distance in optimal transport theory. These schemes allow the numerical approximation of both diffusive and aggregation regimes of different models. Positivity, energy decreasing, and mesh adaptation are built-in in the numerical scheme, and thus we are capable of capturing blow-up densities and of dealing with vacuum regions and merging of mass patches in a natural way.