{"paper":{"title":"Fully Discrete High-Order DG Scheme for Waves: Dispersion and Observability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fully discrete high-order DG schemes for the wave equation exhibit a dispersion-based trapping mechanism leading to exponential blow-up of the observability constant.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Enrique Zuazua, Xiaoyang Wang, Yunzhang Li","submitted_at":"2026-05-17T14:06:53Z","abstract_excerpt":"This paper investigates the spectral structure, numerical dispersion, and observability of fully discrete approximations of the one-dimensional wave equation by $P^k$ (local) discontinuous Galerkin methods. Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order $\\exp(h^{-(1-\\varepsilon)})$ for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary $k$, we propose a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order exp(h^{-(1-ε)}) for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary k, we propose a spectral filtering strategy to restore uniform observability.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper assumes that the fully discrete (space-time) dispersion relation for the P^k DG scheme produces vanishing group velocities at selected frequencies, which is the premise used to derive the exponential blow-up of the observability constant.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Fully discrete DG schemes for the 1D wave equation show dispersion-induced trapping of physical and spurious modes, producing exp(h^{-(1-ε)}) blow-up in observability constants; spectral filtering restores uniform observability, with higher-order methods reducing the filtering burden.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fully discrete high-order DG schemes for the wave equation exhibit a dispersion-based trapping mechanism leading to exponential blow-up of the observability constant.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ecbcba307db0b576b5a1aedeb6cc2d8482b8a48dd639be88e4098b5682da0cac"},"source":{"id":"2605.17464","kind":"arxiv","version":1},"verdict":{"id":"16456a31-c3f0-47b1-bef2-a2088a1298fd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:52:44.750459Z","strongest_claim":"Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order exp(h^{-(1-ε)}) for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary k, we propose a spectral filtering strategy to restore uniform observability.","one_line_summary":"Fully discrete DG schemes for the 1D wave equation show dispersion-induced trapping of physical and spurious modes, producing exp(h^{-(1-ε)}) blow-up in observability constants; spectral filtering restores uniform observability, with higher-order methods reducing the filtering burden.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The paper assumes that the fully discrete (space-time) dispersion relation for the P^k DG scheme produces vanishing group velocities at selected frequencies, which is the premise used to derive the exponential blow-up of the observability constant.","pith_extraction_headline":"Fully discrete high-order DG schemes for the wave equation exhibit a dispersion-based trapping mechanism leading to exponential blow-up of the observability constant."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17464/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.557550Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:01:10.637011Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.702862Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.658269Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"91961e75e9d47f1b4c1b6dd9055710b8731e4118294dc353f9205a1a5a7ecdfe"},"references":{"count":37,"sample":[{"doi":"","year":1995,"title":"1995 , PAGES =","work_id":"1ced35fd-b4f4-4858-a802-1baad8138a8e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"2014 , PAGES =","work_id":"008d57ee-fecd-4e8d-8fd7-248449c99da4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Zuazua, Enrique , TITLE =. 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