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Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion — arXiv:2609.12506

Source

arXiv (en)

Original

https://arxiv.org/abs/2609.12506

License (as archived)

CC0-1.0

Archived

2026-10-16T03:49:04.280Z

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ab65f348769e265c6f1e461e3abc1d1f1af183085aa97e27f9d8f6870830bb48

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Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion — arXiv:2609.12506 — arXiv (en), https://arxiv.org/abs/2609.12506 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ceea8153791c (Metraipolis archive dc_ceea8153791c, sha256 ab65f348769e265c6f1e461e3abc1d1f1af183085aa97e27f9d8f6870830bb48)

@misc{dcceea8153791c,
  title = {Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion — arXiv:2609.12506},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.12506},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ceea8153791c; sha256 ab65f348769e265c6f1e461e3abc1d1f1af183085aa97e27f9d8f6870830bb48},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.12506. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\delta=b_0\tau$, where $\delta$ is the stabilisation parameter, $\tau$ 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 $O(h^{k+1/2}+\tau^\nu)$ when $\delta$ is proportional to $h$ and $\tau=O(h)$, where $h$ is the mesh size, $k$ the polynomial degree, and $\nu$ the temporal order. Stability is established for the $\theta$-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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-14 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.12506 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T03:16:24.994Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-14?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).