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Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes

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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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Acknowledgements

This work was conducted during a visit of F. Rossi to Rutgers University, Camden, NJ, USA. He thanks the institution for its hospitality.

The authors thank the anonymous reviewer for remarks and the suggested bibliography.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rutgers University—Camden, Camden, NJ, USA

    Benedetto Piccoli

  2. LSIS, Aix-Marseille Univ., 13013, Marseille, France

    Francesco Rossi

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  1. Benedetto Piccoli
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  2. Francesco Rossi
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Correspondence to Francesco Rossi.

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Piccoli, B., Rossi, F. Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes. Acta Appl Math 124, 73–105 (2013). https://doi.org/10.1007/s10440-012-9771-6

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  • DOI: https://doi.org/10.1007/s10440-012-9771-6

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