Numerical methods for one-dimensional aggregation equations

TL;DR

This paper develops and analyzes numerical schemes for one-dimensional aggregation equations, ensuring convergence to duality solutions and capturing blow-up phenomena, with applications to kinetic systems in hyperbolic scaling.

Contribution

It introduces a novel discretization method that guarantees convergence to duality solutions and proposes an asymptotic preserving scheme for related kinetic models.

Findings

The discretization converges to duality solutions of the aggregation equation.

The scheme accurately captures finite time blow-up of solutions.

Numerical simulations validate the theoretical convergence and behavior.

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.