Transport equation with nonlocal velocity in wasserstein spaces: Convergence of numerical schemes

Benedetto Piccoli, Francesco Rossi

Research output: Contribution to journalArticlepeer-review

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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.

Original languageEnglish (US)
Pages (from-to)73-105
Number of pages33
JournalActa Applicandae Mathematicae
Volume124
Issue number1
DOIs
StatePublished - Apr 2013

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Keywords

  • Evolution of measures Wasserstein distance
  • Numerical schemes for PDEs
  • Pedestrian modelling
  • Transport equation

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