{
  "abstract": "The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar $r(t)=\\sqrt{\\mathcal{E}_{_\\mathcal{N}}[ϕ]+C}>0$, thereby yielding linear, unconditionally energy-stable schemes for gradient flows. At the discrete level, however, the standard backward Euler and Crank--Nicolson discretizations provide no guarantee that the computed $r^{n+1}$ remains positive, an inconsistency with the continuous definition that contradicts the square-root ansatz and may compromise long-time robustness. Although the SAV method has been widely applied, this subtle but consequential issue has received little attention. We first characterize this failure quantitatively by deriving a sharp criterion and a sufficient condition on the time step size, and construct an explicit counterexample showing that sign loss occurs for parameters of practical relevance. Rather than modifying the definition of $r$ as in existing positivity-preserving variants, we retain the square-root form and reformulate the discrete evolution from $r_t$ to $(r^{2})_t$, which converts the scalar equation into a convex quadratic with a strictly negative constant term, always yielding a unique positive root. For the Crank--Nicolson scheme, the product-form discretization $r^{n+1}r^{n}$ preserves this quadratic structure, while conventional alternatives do not. The resulting schemes incur the same computational cost as the original SAV method and are proved unconditionally energy-stable. Numerical experiments for the Cahn--Hilliard equation confirm the predicted positivity, energy stability, and convergence rates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01063",
  "author_total": 4,
  "authors": [
    {
      "name": "Qiong-Ao Huang"
    },
    {
      "name": "Zhi-Hao Liu"
    },
    {
      "name": "Ying-Wei Wang"
    },
    {
      "name": "Li-Na Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0d5mwekdb6bc8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01063",
  "fulltext_url": null,
  "id": "0d5mwekdb6bc8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01063",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7215760347",
  "pdf_url": "https://arxiv.org/pdf/2610.01063",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0d5mwekdb6bc8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01063",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01063"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation",
  "title_text": "Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/0d5mwekdb6bc8",
  "venue": "arXiv",
  "versions": []
}