Numerical methods for one-dimensional aggregation equations

We focus in this work on the numerical discretization of the one dimensional aggregation equation ${\text{pa}}_t\rho +{\text{pa}}_x(v\rho )=0$, $v=a(W^{\mathrm{\prime }}\ast \rho )$, 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 hydrodynamical limit. Finally numerical simulations are provided to illustrate the results.