{
  "id": "1106.2555",
  "version": "v3",
  "published": "2011-06-13T20:02:51.000Z",
  "updated": "2012-06-05T02:09:55.000Z",
  "title": "Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes",
  "authors": [
    "Benedetto Piccoli",
    "Francesco Rossi"
  ],
  "categories": [
    "math.AP",
    "math.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-06-05T02:09:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35F25"
    ],
    "keywords": [
      "transport equation",
      "wasserstein space",
      "numerical schemes",
      "nonlocal velocity",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.2555P"
    }
  }
}