{
  "abstract": "We propose a second-order, semi-implicit time-stepping scheme for the shallow-water equations on the sphere discretised with compatible finite elements. The scheme splits the advection operator from the wave operator, applying a DG-upwind SSPRK3 method to the advection part and an $H(\\text{div})$-conforming implicit discretisation to the wave part, solved efficiently via hybridisation. The scheme follows a three-step symmetric splitting: an explicit half-step, a full advection step, and an implicit half-step that simultaneously projects the velocity back into the $H(\\text{div})$ space. This symmetry guarantees global second-order accuracy and confines the stability restriction to the CFL condition of the advection step alone. The time-step can therefore be extended beyond this limit by subcycling the advection step without additional implicit solves. The method is implemented in Firedrake and validated against a standard suite of spherical shallow-water benchmarks, demonstrating stable results for Courant numbers up to 1 (with two advection subcycles) with second-order finite elements, second-order convergence, and good parallel scaling.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08169",
  "author_total": 2,
  "authors": [
    {
      "name": "Karina Kowalczyk"
    },
    {
      "name": "Colin Cotter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0b4kgwycvxb2w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08169",
  "fulltext_url": null,
  "id": "0b4kgwycvxb2w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08169",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08169",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0b4kgwycvxb2w.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08169",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08169"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Second-Order Semi-Implicit Projection Time-Stepping Scheme for the Shallow-Water Equations on the Sphere Using Compatible Finite Elements",
  "title_text": "A Second-Order Semi-Implicit Projection Time-Stepping Scheme for the Shallow-Water Equations on the Sphere Using Compatible Finite Elements",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/0b4kgwycvxb2w",
  "venue": "arXiv",
  "versions": []
}