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

2011/06/13 by Benedetto Piccoli, Piccoli, Benedetto, Francesco Rossi +1 · 1 citation
Engineering · Mathematics · #35F25 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1106.2555

openalex publication_date 2011/06/13 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Motivated by pedestrian modelling, we study evolution of measures in the\nWasserstein space. In particular, we consider the Cauchy problem for a\ntransport equation, where the velocity field depends on the measure itself.\n We deal with numerical schemes for this problem and prove convergence of a\nLagrangian scheme to the solution, when the discretization parameters approach\nzero. We also prove convergence of an Eulerian scheme, under more strict\nhypotheses. Both schemes are discretizations of the push-forward formula\ndefined by the transport equation. As a by-product, we obtain existence and\nuniqueness of the solution.\n All the results of convergence are proved with respect to the Wasserstein\ndistance. We also show that L¹ spaces are not natural for such equations,\nsince we lose uniqueness of the solution.\n

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