Convergence analysis of upwind type schemes for the aggregation equation with pointy potential
Annales Henri Lebesgue, Volume 3 (2020) , pp. 217-260.

KeywordsAggregation equation, upwind finite volume scheme, convergence order, measure-valued solution.

### Abstract

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order $1/2$ in Wasserstein distance. The paper is illustrated by numerical simulations that indicate that this convergence order should be optimal.

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