{
  "abstract": "We extend the phi-FD finite-difference scheme, which handles domains described by a level-set function on a Cartesian grid, to Neumann boundary conditions. The condition is relaxed near the boundary through the level-set and an auxiliary field, which is eliminated by writing the relaxed condition at a boundary node and at its nearest interior node. The scheme uses only interior nodes and the level-set: no ghost values, flux reconstruction or cut-cell volumes. We prove optimal consistency and show second-order convergence in the L2, H1 and Linf norms on a non-convex domain. Two elementary precautions, an individual normalization of the boundary equations and a rule for the choice of the coupled interior node, make the condition number O(h^-2) and independent of the smallest cut cell, without any stabilization term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28313",
  "author_total": 5,
  "authors": [
    {
      "name": "Michel Duprez"
    },
    {
      "name": "Vanessa Lleras"
    },
    {
      "name": "Alexei Lozinski"
    },
    {
      "name": "Vincent Vigon"
    },
    {
      "name": "Lisl Weynans"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2e6pp6j9kz7pr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28313",
  "fulltext_url": null,
  "id": "2e6pp6j9kz7pr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28313",
  "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",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214230379",
  "pdf_url": "https://arxiv.org/pdf/2609.28313",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2e6pp6j9kz7pr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28313",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28313"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case",
  "title_text": "phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2e6pp6j9kz7pr",
  "venue": "arXiv",
  "versions": []
}