NUMERICAL METHODS FOR ONE-DIMENSIONAL AGGREGATION EQUATIONS

摘要

We focus in this work on the numerical discretization of the one-dimensional aggregation equation partial derivative(t)rho + partial derivative 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.