Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes |
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Authors: | Benedetto Piccoli Francesco Rossi |
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Institution: | 1. Department of Mathematical Sciences, Rutgers University—Camden, Camden, NJ, USA 2. LSIS, Aix-Marseille Univ., 13013, Marseille, France
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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. |
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