{
  "abstract": "We construct a second-order, linear, and fully decoupled scheme for Rayleigh--Bénard convection between Navier-slip walls. The nonlinear terms are treated explicitly and multiplied by a scalar with exact value $1$ and no energy contributed to the continuous problem. As a result, each BDF2 step, with a rotational pressure correction, solves only constant-coefficient linear problems. For the time discretization, we prove that, for any time-step, the energy decays when the walls have equal temperatures, under a condition on the thermal weight, and the energy is bounded on finite time intervals when heated from below. Relaxing the scalar towards $1$ preserves both estimates. Energy stability does not control the accuracy of the scalar; the velocity and temperature fields are computed with its unrelaxed value, whose deviation from $1$ scales like $(E/C)(δt/τ)^3$ for kinetic energy $E$, time-scale $τ$ of the flow, and weight $C$ of the scalar equation. In our tests, choosing $C$ comparable to $E$ keeps this deviation small and doubles the largest time-step for which steady rolls are reproduced to a relative difference of $10^{-8}$. With Taylor--Hood elements, we observe second-order convergence in time, recover the critical Rayleigh numbers, and match published Nusselt numbers to within $0.5\\%$. A three-dimensional implementation passes corresponding validation tests.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31364",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew McCurdy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqpk21e0fs3ff.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31364",
  "fulltext_url": null,
  "id": "bqpk21e0fs3ff",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31364",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.31364",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqpk21e0fs3ff.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.31364",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31364"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection",
  "title_text": "A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bqpk21e0fs3ff",
  "venue": "arXiv",
  "versions": []
}