{"paper":{"title":"Low Stage High Order Explicit Runge--Kutta Methods via Q- and D-Conditions: General Theory and Efficient Recursive Construction","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A Q/D-space reformulation of order conditions yields explicit Runge-Kutta methods of even order p with stage count (p squared minus 2p plus 8) over 4.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jizu Huang, Junyuan He","submitted_at":"2026-05-16T13:43:57Z","abstract_excerpt":"Constructing explicit Runge--Kutta (ERK) methods with as few stages as possible for a given order is a classical problem in numerical analysis. In this work, we introduce a $Q$/$D$-space framework of sufficient order conditions for ERK methods. This framework generalizes Butcher's classical simplifying assumptions by reformulating them in terms of simplified $Q$- and $D$-spaces defined through their residual vectors. It yields sufficient conditions which, together with $B(p)$, ensure order $p$. It also leads to a recursive construction procedure for ERK methods of arbitrary even order, in whic"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every even order p ≥ 4 the construction produces explicit Runge-Kutta methods with stage number s(p) = (p² - 2p + 8)/4; the Q/D conditions together with B(p) are sufficient to guarantee order p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Q- and D-space residual conditions remain linearly independent and the two structured linear systems at each recursive step remain solvable for the chosen stage count; this is invoked when the authors state that the Butcher coefficients are obtained from the two linear systems (abstract and §3-4).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Q/D-space reformulation of Butcher simplifying assumptions yields sufficient order conditions and a recursive linear-system construction for explicit Runge-Kutta methods of even order p with s(p)=(p²-2p+8)/4 stages.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Q/D-space reformulation of order conditions yields explicit Runge-Kutta methods of even order p with stage count (p squared minus 2p plus 8) over 4.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"831c4c4a95b4b267bf71b4956d8c0c92f4b9927313fabbc14cc13e07a3bceffc"},"source":{"id":"2605.16995","kind":"arxiv","version":1},"verdict":{"id":"bc09107e-a7cd-46b6-988f-97d5e1aa5a20","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:10:04.663515Z","strongest_claim":"For every even order p ≥ 4 the construction produces explicit Runge-Kutta methods with stage number s(p) = (p² - 2p + 8)/4; the Q/D conditions together with B(p) are sufficient to guarantee order p.","one_line_summary":"A Q/D-space reformulation of Butcher simplifying assumptions yields sufficient order conditions and a recursive linear-system construction for explicit Runge-Kutta methods of even order p with s(p)=(p²-2p+8)/4 stages.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Q- and D-space residual conditions remain linearly independent and the two structured linear systems at each recursive step remain solvable for the chosen stage count; this is invoked when the authors state that the Butcher coefficients are obtained from the two linear systems (abstract and §3-4).","pith_extraction_headline":"A Q/D-space reformulation of order conditions yields explicit Runge-Kutta methods of even order p with stage count (p squared minus 2p plus 8) over 4."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16995/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"citation_quote_validity","ran_at":"2026-05-19T19:50:02.541722Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:18.903645Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T19:23:36.219732Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:20:41.580946Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.203358Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.292561Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"11084f1a2023915264a62f51cbf3fd6a69dcb725119e6b32614e4bb0267de02a"},"references":{"count":38,"sample":[{"doi":"","year":2016,"title":"Tensor networks and hi- erarchical tensors for the solution of high-dimensional partial differential equations","work_id":"d76da37e-686c-42a4-b34f-1d755f4e8335","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Counting Rooted Trees: The Universal Law t(n) ∼ Cρ −nn−3/2","work_id":"3ea2841b-af68-4a37-b34f-14e426b230ac","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"A 3(2) pair of Runge-Kutta formulas","work_id":"f7f109bd-db00-4b17-87fc-e70545239090","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1963,"title":"Coefficients for the study of Runge-Kutta integration processes","work_id":"5f755197-c7e6-41fa-ac35-0e57a0ad145a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1964,"title":"Implicit runge-kutta processes","work_id":"e50396ab-86cd-429e-a810-3c13ce531974","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"c7a439d78e520a819f3e184cc32c4fc9e49ffec5e4484afceb6fb3734d10912b","internal_anchors":2},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ea07115783fc62ad7578425c7976b92e521272d5338a3ff29e24f845dd6ef11f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}