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. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.
Highlights
This paper is devoted to the numerical approximation of measure valued solutions to the so-called aggregation equation in space dimension d
W plays the role of an interaction potential whose gradient ∇xW (x − y) measures the relative force exerted by a unit mass localized at a point y onto a unit mass located at a point x. This system appears in many applications in physics and population dynamics
The most frequently used model is the Othmer–Dunbar–Alt system, the hydrodynamic limit of which leads to the aggregation equation (1.1), see [DS05, FLP05, JV13]
Summary
This paper is devoted to the numerical approximation of measure valued solutions to the so-called aggregation equation in space dimension d. For a given velocity field, the study of the order of convergence for the finite volume upwind scheme for the transport equation has received a lot of attention. This scheme is known to be first order convergent in L∞ norm for any smooth initial data in C2(Rd) and for well-suited meshes, provided a standard stability condition (Courant–Friedrichs–Lewy condition) holds, see [BGP05]. We adapt the strategy initiated in [DLV17] to prove the convergence at order 1/2 of an upwind scheme for the aggregation equation for which the velocity field depends on the solution in a nonlinear way. We show that the order of convergence is optimal and we provide several numerical simulations in which we recover the behavior of the solutions after blow-up time
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