{
  "abstract": "We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential $u$ and density $a$ solving $-\\mathrm{div}(a\\nabla u) = f^+-f^-$, via $\\dot T = -a\\nabla u(T)/[(1-t)f^+(T)+tf^-(T)]$, $T(0,x)=x$. In 1D this gives a closed form. In 2D--left unfinished in an earlier report--we obtain $(u,a)$ via a convex Beckmann flow problem (second-order-cone programming), cross-checked against a dual LP, then build the map by direct integration or a ray-based 1D \"clock\" ODE with explicit density per ray. Validation uses strong duality, mass conservation, and vanishing-regularization limits, on a translation-invariant case and two 2D examples (a radial cone, a checkerboard) checked by symmetry; both show boundary-concentrated mass and non-unique flux where supports touch. We extend to 3D (theory unchanged), matching the 2D answer to six digits and reproducing the same phenomena on a radial ball, exposing a limit of the 2D visualization style. Movies illustrating the transport maps are available as ancillary files on the arXiv page of this submission.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29987",
  "author_total": 2,
  "authors": [
    {
      "name": "Babacar Mbaye Ndiaye"
    },
    {
      "name": "Diaraf Seck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ep4myjv65kvr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29987",
  "fulltext_url": null,
  "id": "ep4myjv65kvr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29987",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214401183",
  "pdf_url": "https://arxiv.org/pdf/2609.29987",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ep4myjv65kvr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29987",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29987"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions",
  "title_text": "A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/ep4myjv65kvr9",
  "venue": "arXiv",
  "versions": []
}