{
  "abstract": "We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton--Jacobi--Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is \\(O(τΔt^{γ/2})\\), and the total mass of interpolated branches is at most \\(2τ/Δt\\), where \\(τ\\) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as \\(τ\\to0\\). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield \\[ \\|(u-u_h)^+\\|_\\infty\\le C(Δt^{1/4}+hΔt^{-1/2}), \\qquad \\|(u_h-u)^+\\|_\\infty\\le C(Δt^{1/10}+h^{1/2}Δt^{-1/4}). \\] Thus \\(Δt\\asymp h^{10/7}\\) gives an \\(O(h^{1/7})\\) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17186",
  "author_total": 2,
  "authors": [
    {
      "name": "Haoran Xu"
    },
    {
      "name": "Xingye Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2egp6py6ve7wx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17186",
  "fulltext_url": null,
  "id": "2egp6py6ve7wx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17186",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17186",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2egp6py6ve7wx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17186",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17186"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations",
  "title_text": "Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/2egp6py6ve7wx",
  "venue": "arXiv",
  "versions": []
}