{
  "abstract": "Stable discrete approximation of wave controls requires observability estimates that remain uniform under mesh refinement. Such uniformity is known to fail for many low-order discretizations, but the quantitative understanding of observability loss for arbitrary high-order approximations is largely open, owing to their multiple spectral branches and the absence of explicit dispersion relations. To address this gap, we establish a quantitative theory of observability loss and recovery for fully discrete $Q^k$ local discontinuous Galerkin approximations of the multidimensional wave equation, valid for any fixed polynomial degree $k$. Our analysis of the $(k+1)^d$-branch discrete spectrum identifies a unique physical branch that recovers the continuous propagation speed at low frequencies. Under a strict stability margin on this branch, its group velocity vanishes at high-frequency symmetry points. To quantify the resulting loss of uniform observability, we prove that the observability constant satisfies an exponential lower bound of the form $\\exp(Ch^{-(1-\\varepsilon)})$ for any $\\varepsilon\\in(0,1)$ by constructing Gevrey-localized wave packets. We then provide a recovery strategy that restores uniform observability for any fixed polynomial degree $k$, with a sufficient observation time controlled by the minimum retained physical group velocity. Finally, our numerical results suggest that higher-order approximations retain a broader genuine physical-frequency band at a prescribed positive group-speed threshold, allowing less restrictive filtering in wave-control computation and better preserving wave dynamics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.17464",
  "author_total": 2,
  "authors": [
    {
      "name": "Yunzhang Li"
    },
    {
      "name": "Xiaoyang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cpcy1q3w3ws0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.17464",
  "fulltext_url": null,
  "id": "5cpcy1q3w3ws0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.17464",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7161712768",
  "pdf_url": "https://arxiv.org/pdf/2605.17464",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cpcy1q3w3ws0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.17464",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.17464"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery",
  "title_text": "Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5cpcy1q3w3ws0",
  "venue": "arXiv",
  "versions": []
}