{
  "abstract": "We analyse fully discrete Streamline Diffusion finite element methods, also known as SUPG methods, for time-dependent first-order systems with skew-symmetric spatial operators. To the best of our knowledge, this is the first stability and a priori error analysis of strongly consistent Streamline Diffusion finite elements combined with linear multistep schemes of order greater than two for multidimensional first-order hyperbolic systems. Local-in-time $L^2$-residual minimisation singles out $δ=b_0τ$, where $δ$ is the stabilisation parameter, $τ$ the time step, and $b_0$ the current-time coefficient in the method average. For this choice, each implicit linear system is a symmetric positive definite normal equation in a graph norm. A direct dual-residual estimate yields errors of order $O(h^{k+1/2}+τ^ν)$ when $δ$ is proportional to $h$ and $τ=O(h)$, where $h$ is the mesh size, $k$ the polynomial degree, and $ν$ the temporal order. Stability is established for the $θ$-method and the Adams-Moulton methods AM3-AM5. The general-degree AM3 and AM4 estimates use strengthened CFL conditions; for continuous piecewise affine elements, an elementwise cancellation recovers the standard hyperbolic regime. AM5 and the explicit Adams-Bashforth methods AB3 and AB4 are stable under a standard hyperbolic CFL condition, though the explicit schemes do not have normal-equation structure. Order-matched acoustic tests recover orders two through five and material-residual rates one half-order lower. In a d'Alembert test with discontinuous initial data, normal-equation SUPG recovers second-order local $\\mathbb P_1$ convergence away from the wave fronts, whereas the unstabilised error is close to half order. A two-dimensional compact-wave test shows improved localisation with modest dissipation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12506",
  "author_total": 3,
  "authors": [
    {
      "name": "Erik Burman"
    },
    {
      "name": "Peter Hansbo"
    },
    {
      "name": "Mats G. Larson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/djbb1w1h8g043.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12506",
  "fulltext_url": null,
  "id": "djbb1w1h8g043",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12506",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7212975210",
  "pdf_url": "https://arxiv.org/pdf/2609.12506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/djbb1w1h8g043.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion",
  "title_text": "Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/djbb1w1h8g043",
  "venue": "arXiv",
  "versions": []
}