{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:FVWNZ3OOKYNHUIJBWTBTZH22YM","short_pith_number":"pith:FVWNZ3OO","schema_version":"1.0","canonical_sha256":"2d6cdcedce561a7a2121b4c33c9f5ac323430eb15ff70f9cadd1307bbb1c2bc5","source":{"kind":"arxiv","id":"2108.02910","version":1},"attestation_state":"computed","paper":{"title":"Weighted and shifted BDF2 methods on variable grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Fan Yu, Minghua Chen, Qingdong Zhang","submitted_at":"2021-08-06T01:49:05Z","abstract_excerpt":"Variable steps implicit-explicit multistep methods for PDEs have been presented in [17], where the zero-stability is studied for ODEs; however, the stability analysis still remains an open question for PDEs. Based on the idea of linear multistep methods, we present a simple weighted and shifted BDF2 methods with variable steps for the parabolic problems, which serve as a bridge between BDF2 and Crank-Nicolson scheme. The contributions of this paper are as follows: we first prove that the optimal adjacent time-step ratios for the weighted and shifted BDF2, which greatly improve the maximum time"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2108.02910","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2021-08-06T01:49:05Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b14bc51c33f141f624b32f37d4e7fdc3b42c1bf0bd0c3c2eb218ee4590e52f84","abstract_canon_sha256":"88cfc3f4bc422f20dd20a41ac2372610902841f58a6ada8d45fb4173c02ef301"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:03:43.521536Z","signature_b64":"wUFhF27hMnfVsw+SBMeFgMvInpFA4Oot6r5jGxO8WU8WliV2/bVLPXWMQfhqyFoGnKwtTCZU0P3Te6fhPB4jAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d6cdcedce561a7a2121b4c33c9f5ac323430eb15ff70f9cadd1307bbb1c2bc5","last_reissued_at":"2026-07-05T03:03:43.521076Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:03:43.521076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted and shifted BDF2 methods on variable grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Fan Yu, Minghua Chen, Qingdong Zhang","submitted_at":"2021-08-06T01:49:05Z","abstract_excerpt":"Variable steps implicit-explicit multistep methods for PDEs have been presented in [17], where the zero-stability is studied for ODEs; however, the stability analysis still remains an open question for PDEs. Based on the idea of linear multistep methods, we present a simple weighted and shifted BDF2 methods with variable steps for the parabolic problems, which serve as a bridge between BDF2 and Crank-Nicolson scheme. The contributions of this paper are as follows: we first prove that the optimal adjacent time-step ratios for the weighted and shifted BDF2, which greatly improve the maximum time"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.02910","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.02910/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2108.02910","created_at":"2026-07-05T03:03:43.521136+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.02910v1","created_at":"2026-07-05T03:03:43.521136+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.02910","created_at":"2026-07-05T03:03:43.521136+00:00"},{"alias_kind":"pith_short_12","alias_value":"FVWNZ3OOKYNH","created_at":"2026-07-05T03:03:43.521136+00:00"},{"alias_kind":"pith_short_16","alias_value":"FVWNZ3OOKYNHUIJB","created_at":"2026-07-05T03:03:43.521136+00:00"},{"alias_kind":"pith_short_8","alias_value":"FVWNZ3OO","created_at":"2026-07-05T03:03:43.521136+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM","json":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM.json","graph_json":"https://pith.science/api/pith-number/FVWNZ3OOKYNHUIJBWTBTZH22YM/graph.json","events_json":"https://pith.science/api/pith-number/FVWNZ3OOKYNHUIJBWTBTZH22YM/events.json","paper":"https://pith.science/paper/FVWNZ3OO"},"agent_actions":{"view_html":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM","download_json":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM.json","view_paper":"https://pith.science/paper/FVWNZ3OO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.02910&json=true","fetch_graph":"https://pith.science/api/pith-number/FVWNZ3OOKYNHUIJBWTBTZH22YM/graph.json","fetch_events":"https://pith.science/api/pith-number/FVWNZ3OOKYNHUIJBWTBTZH22YM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM/action/storage_attestation","attest_author":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM/action/author_attestation","sign_citation":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM/action/citation_signature","submit_replication":"https://pith.science/pith/FVWNZ3OOKYNHUIJBWTBTZH22YM/action/replication_record"}},"created_at":"2026-07-05T03:03:43.521136+00:00","updated_at":"2026-07-05T03:03:43.521136+00:00"}