{
  "abstract": "This paper proposes a new class of efficient preconditioners for variable-coefficient time–space fractional diffusion equations. The Caputo time-fractional derivative is discretized by the L2-1σ scheme, while the Riesz space-fractional derivative is approximated by the shifted Grünwald–Letnikov formula, yielding an all-at-once system amenable to parallel solution. Because the coefficient matrix A of the resulting discrete system no longer has a strict Toeplitz structure, solving the system entails substantial storage and computational costs. To address this difficulty, we construct a class of Strang–τ structured preconditioners. The variable-coefficient diagonal matrix is first approximated using the arithmetic mean of the diffusion coefficient; the temporal and spatial Toeplitz matrices are then approximated by a Strang circulant matrix and a τ matrix, respectively. The preconditioner is applied efficiently using the fast Fourier transform (FFT), the discrete sine transform (DST), and the Sherman–Morrison rank-one update formula. We further derive a spectral distribution estimate showing that the eigenvalues of the preconditioned coefficient matrix P˜−1A lie in a disk centered at 1, and its radius is controlled jointly by the temporal low-rank correction and circulant-approximation errors, the spatial τ-approximation error, and the mean-approximation error of the variable coefficient. Numerical experiments demonstrate that the proposed preconditioner substantially accelerates GMRES(m) convergence.",
  "abstract_shown": true,
  "affiliations": [
    "School of Mathematics and Statistics, Beihua University, Jilin 132013, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Jianyu Zhao"
    },
    {
      "affs": [
        0
      ],
      "name": "Meijuan Wang"
    },
    {
      "affs": [
        0
      ],
      "name": "Yi Wang"
    },
    {
      "affs": [
        0
      ],
      "name": "Haifeng Sang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47m889bntbfa3.bib",
  "cited_by": 0,
  "doi": "10.3390/math14213957",
  "fulltext_url": null,
  "id": "47m889bntbfa3",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/3957",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/47m889bntbfa3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14213957",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/3957"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients",
  "title_text": "An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients",
  "type": "peer-reviewed",
  "updated_at": "2026-10-31T00:20:46Z",
  "url": "https://research.draconitic.com/paper/47m889bntbfa3",
  "venue": "Mathematics",
  "versions": []
}