{
  "abstract": "Summation-by-parts (SBP) operators are a cornerstone of provably stable high-order methods for hyperbolic conservation laws. For nonlinear problems, the flux-differencing framework of Fisher and Carpenter combines such SBP operators with entropy-conservative two-point fluxes to obtain semi-discretizations that are conservative in the sense of Lax--Wendroff and discretely entropy-conservative or stable. So far, this framework has been developed exclusively for operators that are exact for polynomials. Function-space SBP (FSBP) operators lift this restriction by being exact on a general finite-dimensional function space that can be tailored to the problem at hand. In this work, we extend flux differencing to FSBP operators for general (potentially non-polynomial) function spaces. We prove that, whenever the operator has a diagonal norm and its function space contains the constants, the resulting semi-discretization is conservative in the sense of Lax--Wendroff, semi-discretely entropy-conservative within a single element, and entropy-stable across element interfaces. We also provide the first theoretical accuracy analysis of FSBP operators: We characterize their design order through an algebraic criterion on the underlying function space, prove that suitable non-polynomial spaces achieve the same design order as polynomial spaces of equal dimension, and show that the flux-differencing semi-discretization retains this order. Notably, the entropy-conservative two-point flux is reused unchanged; only the underlying function space differs from the polynomial case. Numerical experiments for the Burgers' equation and the compressible Euler equations in one and two dimensions confirm these properties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30272",
  "author_total": 3,
  "authors": [
    {
      "name": "Henry M. Haase"
    },
    {
      "name": "Andrew R. Winters"
    },
    {
      "name": "Jan Glaubitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6p504tq1yxv4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30272",
  "fulltext_url": null,
  "id": "a6p504tq1yxv4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30272",
  "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",
      "2206|Computational Fluid Dynamics and Aerodynamics"
    ]
  ],
  "openalex": "W7223307195",
  "pdf_url": "https://arxiv.org/pdf/2610.30272",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6p504tq1yxv4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30272",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30272"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy",
  "title_text": "Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/a6p504tq1yxv4",
  "venue": "arXiv",
  "versions": []
}