Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

François Delarue (1) , Frédéric Lagoutìère (2) , Nicolas Vauchelet (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 MMCS - Modélisation mathématique, calcul scientifique
3 LAGA - Laboratoire Analyse, Géométrie et Applications

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. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.

Mots clés: convergence order; aggregation equation; measure-valued solution; upwind finite volume scheme

Domaines: Equations aux dérivées partielles [math.AP]

François Delarue, Frédéric Lagoutìère, Nicolas Vauchelet. Convergence analysis of upwind type schemes for the aggregation equation with pointy potential. Annales Henri Lebesgue, In press, ⟨10.5802/ahl.30⟩. ⟨hal-01591602v2⟩