{
  "abstract": "We consider global Chebyshev collocation for the ignition problems $y'=y^q(1-y)$, $y(0)=δ$, $q\\ge2$; for $q=2$ this is the flame problem, a standard stiff test equation. After rescaling to $[-1,1]$ the solution has an interior layer of width $O(μ)$, $μ=(q-1)δ^{q-1}$, at a $δ$-dependent position. Two obstacles arise: plain Chebyshev collocation needs $O(1/μ)$ nodes, and imposing the initial condition amplifies errors by a factor of order $1/μ$, which makes the collocation equations severely ill-conditioned. Our main contribution addresses the second one. Since all solutions are translates of one profile, the initial condition can be replaced by an equivalent interior condition. For an unknown $u=φ(y)$, imposing it where the speed $|φ'f_q|$ of $u$ along the orbit is maximal bounds the propagator of the linearized equation by one; for $u=\\ln y$ this is $y(x^\\ast)=(q-1)/q$. For the first obstacle we use the sinh map of Tee and Trefethen: the time function is explicit, the complex singularities of the profile lie exactly at $\\pm(2k+1)πi$, and the map parameters follow in closed form, so that $O(\\ln(1/μ))$ nodes suffice. In double precision the pinned scheme reaches the interpolation error of the exact solution in five or six Newton iterations for $μ$ down to $10^{-12}$, whereas the initial-condition form fails or converges to wrong discrete solutions, also in an integral (Birkhoff) reformulation. Computations in $256$-bit arithmetic confirm the predicted rate down to errors of $10^{-70}$, AAA approximation locates the singularities to within $2\\%$, and the method carries over to an Arrhenius reaction rate, with all parameters computed numerically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27864",
  "author_total": 1,
  "authors": [
    {
      "name": "Imre Boros"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2h600mjhpvxcd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27864",
  "fulltext_url": null,
  "id": "2h600mjhpvxcd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27864",
  "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"
    ],
    [
      "openalex",
      "2612|Differential Equations and Numerical Methods"
    ]
  ],
  "openalex": "W7223159689",
  "pdf_url": "https://arxiv.org/pdf/2610.27864",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2h600mjhpvxcd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27864",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27864"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Front-pinned spectral collocation for the flame problem and related ignition problems",
  "title_text": "Front-pinned spectral collocation for the flame problem and related ignition problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/2h600mjhpvxcd",
  "venue": "arXiv",
  "versions": []
}