Numerical Methods for One-Dimensional Aggregation Equations

Abstract

We focus in this work on the numerical discretization of the one-dimensional aggregation equation $\partial_t\rho + \partial_x (v\rho)=0$, $v=a(W'*\rho)$, in the attractive case. Finite time blow up of smooth initial data occurs for potential $W$ having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in the hydrodynamic limit. Finally numerical simulations are provided to illustrate the results.