{
  "abstract": "We develop and analyze a fully discrete, globally divergence-free hybridizable discontinuous Galerkin (HDG) method for a gradient-based Smagorinsky model. The method combines backward Euler time stepping, interior-penalty discretizations of molecular and nonlinear eddy diffusion, and an upwind convective flux. The discrete velocity is $H(\\operatorname{div})$-conforming and pointwise divergence-free, which yields pressure robustness. For sufficiently large penalty parameters, we prove unconditional energy stability and existence of a discrete solution, and establish uniqueness under additional smallness conditions. A velocity error estimate is derived without explicit inverse powers of the molecular viscosity. The nonlinear facet residuals are controlled using local trace-approximation estimates and a viscosity-independent facet penalty. We retain the dependence of the discrete Gronwall factor on the filter scale and the mesh size; a mesh-uniform bound follows under suitable solution regularity, a fixed time-step margin, and the scaling $δ=O(h)$ on quasi-uniform meshes. The reported manufactured-solution results are consistent with the resulting pre-asymptotic error bounds. Further flow examples illustrate the dissipative behavior of the method and are distinguished from the boundary conditions and parameter range covered by the analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18652",
  "author_total": 2,
  "authors": [
    {
      "name": "Shuaijun Liu"
    },
    {
      "name": "Xiaoping Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/83padygfpc2vv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18652",
  "fulltext_url": null,
  "id": "83padygfpc2vv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18652",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/83padygfpc2vv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model",
  "title_text": "A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/83padygfpc2vv",
  "venue": "arXiv",
  "versions": []
}