{"paper":{"title":"Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Specular differentiation defines nonlinear numerical schemes for first-order ODE initial value problems with proven second-order accuracy and exact integration along ellipses.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Kiyuob Jung","submitted_at":"2026-01-14T22:14:24Z","abstract_excerpt":"This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The quasi-Fermat theorem and quasi-Mean Value Theorem hold for the newly defined specular differentiation operator and can be used to derive the truncation error bounds without additional restrictions on the solution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces specular differentiation to create second-order consistent numerical schemes for ODE IVPs, including a modified version with zero local truncation error on elliptical trajectories.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Specular differentiation defines nonlinear numerical schemes for first-order ODE initial value problems with proven second-order accuracy and exact integration along ellipses.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a07c47602e617c9bb686567b0c697c20b8cef0138136fea1affe26899b466ab8"},"source":{"id":"2601.09900","kind":"arxiv","version":4},"verdict":{"id":"d4c514f6-505c-478a-bf55-efeb92c0e203","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:48:00.327433Z","strongest_claim":"We select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.","one_line_summary":"Introduces specular differentiation to create second-order consistent numerical schemes for ODE IVPs, including a modified version with zero local truncation error on elliptical trajectories.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The quasi-Fermat theorem and quasi-Mean Value Theorem hold for the newly defined specular differentiation operator and can be used to derive the truncation error bounds without additional restrictions on the solution.","pith_extraction_headline":"Specular differentiation defines nonlinear numerical schemes for first-order ODE initial value problems with proven second-order accuracy and exact integration along ellipses."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.09900/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}