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

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.

Dans cet article, nous proposons une analyse numérique de schémas de type upwind pour l’équation d’agrégation nonlocale et nonlinéaire. Dans cette approche, l’équation d’agrégation est interprétée comme une équation de transport conservative avec un champs de vitesse nonlocal et nonlinéaire de faible régularité. En particulier, le potentiel d’interaction peut être pointu, dans ce cas le champs de vitesse peut avoir des discontinuités. En se basant sur des résultats récents d’existence et d’unicité d’un flot de Filippov pour ce type d’équations, nous étudions un schéma volume fini de type upwind et nous montrons qu’il converge à l’ordre 1/2 en distance de Wasserstein. Ce résultat est illustré par des simulations numériques indiquant que cet ordre de convergence est optimal.

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DOI: 10.5802/ahl.30
Classification: 35B40, 35D30, 35L60, 35Q92, 49K20
Keywords: Aggregation equation, upwind finite volume scheme, convergence order, measure-valued solution.
Delarue, François 1; Lagoutière, Frédéric 2; Vauchelet, Nicolas 3

1 Université Côte d’Azur, Laboratoire J.A. Dieudonné, CNRS UMR 7351, 06108 Nice cedex 02, France
2 Université Sorbonne Paris Nord, Laboratoire Analyse, Géométrie et Applications, LAGA, CNRS UMR 7539, 93430, Villetaneuse, France
3 Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse Géométrie et Applications, 93430 Villetaneuse, France
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Delarue, François; Lagoutière, Frédéric; Vauchelet, Nicolas. Convergence analysis of upwind type schemes for the aggregation equation with pointy potential. Annales Henri Lebesgue, Volume 3 (2020), pp. 217-260. doi : 10.5802/ahl.30. http://archive.numdam.org/articles/10.5802/ahl.30/

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