{
  "abstract": "Gradient random walk (GRW) methods represent a solution's spatial derivative by weighted particles and recover the solution by cumulative summation. We examine their accuracy for the one-dimensional heat, scalar cubic reaction-diffusion, and Burgers' equations. Thirty-seed ensembles, paired reconstructions, and deterministic controls identify the sources of two apparent accuracy limits. For the heat equation, comparing cumulative bin sums at bin centers introduces a first-order alignment error, which the exact reflected-particle distribution predicts and bin-edge comparison removes. An identity for the expected squared error predicts the remaining floor and the particle count for a target root-mean-square accuracy. With unequal weights, a count chosen in advance gives an error within 1% of its prediction. Reaction-front position, speed, and shape errors decrease under refinement. For Burgers' equation, exact transformed boundary data reduce the deterministic Cole-Hopf error about 490-fold, while fixed-endpoint particle runs approach their deterministic reference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22592",
  "author_total": 2,
  "authors": [
    {
      "name": "Stephen Abkin"
    },
    {
      "name": "Prabir Daripa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0qb7eqbg11qf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22592",
  "fulltext_url": null,
  "id": "e0qb7eqbg11qf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22592",
  "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",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7204164990",
  "pdf_url": "https://arxiv.org/pdf/2608.22592",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0qb7eqbg11qf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2608.22592",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22592"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Error attribution in gradient random walk methods for parabolic equations",
  "title_text": "Error attribution in gradient random walk methods for parabolic equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/e0qb7eqbg11qf",
  "venue": "arXiv",
  "versions": []
}