Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes

TL;DR

This paper investigates the convergence of numerical schemes for a measure-valued transport equation with nonlocal velocity in Wasserstein spaces, motivated by pedestrian modeling, establishing convergence, existence, and uniqueness results.

Contribution

It introduces and proves the convergence of both Lagrangian and Eulerian numerical schemes for a measure-dependent transport equation in Wasserstein spaces, with new theoretical guarantees.

Findings

Lagrangian scheme converges to the solution as discretization parameters approach zero.

Eulerian scheme converges under stricter hypotheses.

Existence and uniqueness of solutions are established in Wasserstein spaces.

Abstract

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that $L^1$ spaces are not natural for such equations, since we lose uniqueness of the solution.