{
  "abstract": "We develop an adaptive time-stepping strategy for scalar auxiliary variable (SAV) schemes combined with variable-step BDF2 discretization for gradient flows. The strategy selects step sizes based on the instantaneous energy dissipation rate and enforces monotonicity of a computable symmetric energy through a step rejection mechanism. We show that the step-size rules used in practice correspond to equidistributing a monitor function along the energy curve, connecting energy-adaptive time stepping to the classical theory of r-adaptive mesh methods. The monitor $g = |\\dot{E}|$ is asymptotically optimal as it minimises the leading-order maximum symmetric-energy discrepancy for a given number of steps. A combined monitor that switches to $g = \\sqrt{|\\dot{E}|}$ near equilibrium caps step growth without degrading this error bound. The arclength parameter $β$ used in prior work is shown to be the ratio of the largest permitted step to the energy-error tolerance. The controller is paired with the SSAV--BDF2 discretization. Accepted steps satisfy monotonicity of the symmetric energy, from which we deduce a bound on the physical energy that is uniform in the number of steps, $\\sup_n E(u^n) \\le E(u^{0}) + τ_1 + O(Δt_{\\max}^2)$, and the scheme achieves second-order global accuracy under standard step-ratio restrictions. Numerical experiments on Cahn--Hilliard, Ohta--Kawasaki, Phase--Field Crystal, and functionalized Cahn--Hilliard models confirm the theory and demonstrate the method's efficiency.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30454",
  "author_total": 2,
  "authors": [
    {
      "name": "Suki Shergill"
    },
    {
      "name": "JF Williams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bb54ng7ypcbt0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30454",
  "fulltext_url": null,
  "id": "bb54ng7ypcbt0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30454",
  "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",
      "2612|Numerical methods for differential equations"
    ]
  ],
  "openalex": "W7223294524",
  "pdf_url": "https://arxiv.org/pdf/2610.30454",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bb54ng7ypcbt0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30454",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30454"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows",
  "title_text": "Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bb54ng7ypcbt0",
  "venue": "arXiv",
  "versions": []
}