  We focus in this work on the numerical discretization of the one dimensional
aggregation equation $\pa_t\rho + \pa_x (v\rho)=0$, $v=a(W'*\rho)$, 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 hydrodynamical limit. Finally numerical simulations are
provided to illustrate the results.
