Numerical methods for one-dimensional aggregation equations

Abstract : 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.
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SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2015, 53 (2), pp.895-916. 〈10.1137/140959997〉
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Francois James, Nicolas Vauchelet. Numerical methods for one-dimensional aggregation equations. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2015, 53 (2), pp.895-916. 〈10.1137/140959997〉. 〈hal-00955971v2〉

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