{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:2ADBBFQ5OII5JMZBMD24SBUGH7","short_pith_number":"pith:2ADBBFQ5","canonical_record":{"source":{"id":"1908.08732","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-23T09:35:31Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"84e393cdaa0e72c082a235d1d3c4a37341fc3346c6231f288785e28318be36a3","abstract_canon_sha256":"63cbbaa8c46d765c3d5cf442da30ad7ed78c9583182a74f7dd163dd984a22c08"},"schema_version":"1.0"},"canonical_sha256":"d00610961d7211d4b32160f5c906863ffffd7343185c5a9c9ce05294dbff7738","source":{"kind":"arxiv","id":"1908.08732","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08732","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08732v2","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08732","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"2ADBBFQ5OII5","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_16","alias_value":"2ADBBFQ5OII5JMZB","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_8","alias_value":"2ADBBFQ5","created_at":"2026-07-05T00:39:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:2ADBBFQ5OII5JMZBMD24SBUGH7","target":"record","payload":{"canonical_record":{"source":{"id":"1908.08732","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-23T09:35:31Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"84e393cdaa0e72c082a235d1d3c4a37341fc3346c6231f288785e28318be36a3","abstract_canon_sha256":"63cbbaa8c46d765c3d5cf442da30ad7ed78c9583182a74f7dd163dd984a22c08"},"schema_version":"1.0"},"canonical_sha256":"d00610961d7211d4b32160f5c906863ffffd7343185c5a9c9ce05294dbff7738","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:39:32.303739Z","signature_b64":"FKqFyKvXNjsa4Ou8kQ9Ua1FhoOrpZdx9XC9NJO9Ffn+JF2yZkedExItxfhFUtljMR/EKORygrt3PKZ/Wy5jiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d00610961d7211d4b32160f5c906863ffffd7343185c5a9c9ce05294dbff7738","last_reissued_at":"2026-07-05T00:39:32.303247Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:39:32.303247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.08732","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bacvA0LJH/bvSstYd9xc6DnE9yafjvmBpMqqKWQkNAu7KuGt8DHrcIAvV2OUDLiVCNRC2dKDvwR4OAYhq+BiCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T19:00:58.765213Z"},"content_sha256":"2b9f4208ba8c792429baaa5b82c4e57440f3ad3074de22374dd87bf73c3dab0a","schema_version":"1.0","event_id":"sha256:2b9f4208ba8c792429baaa5b82c4e57440f3ad3074de22374dd87bf73c3dab0a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:2ADBBFQ5OII5JMZBMD24SBUGH7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch","submitted_at":"2019-08-23T09:35:31Z","abstract_excerpt":"In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated to grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08732","kind":"arxiv","version":2},"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/1908.08732/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ID2EmLN+f/VWUMzk6p+3ABzaLWDTuiWYlag8rx+L3dKj3mYRf090MKhdwcjgJaZMeQ5slP4uXIohc6H8rERsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T19:00:58.765699Z"},"content_sha256":"2c3d7687c1c2468212e7451141dae05cd2ec8a07ab35cd697c52c9220424de14","schema_version":"1.0","event_id":"sha256:2c3d7687c1c2468212e7451141dae05cd2ec8a07ab35cd697c52c9220424de14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/bundle.json","state_url":"https://pith.science/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-14T19:00:58Z","links":{"resolver":"https://pith.science/pith/2ADBBFQ5OII5JMZBMD24SBUGH7","bundle":"https://pith.science/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/bundle.json","state":"https://pith.science/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2ADBBFQ5OII5JMZBMD24SBUGH7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2ADBBFQ5OII5JMZBMD24SBUGH7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"63cbbaa8c46d765c3d5cf442da30ad7ed78c9583182a74f7dd163dd984a22c08","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-23T09:35:31Z","title_canon_sha256":"84e393cdaa0e72c082a235d1d3c4a37341fc3346c6231f288785e28318be36a3"},"schema_version":"1.0","source":{"id":"1908.08732","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08732","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08732v2","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08732","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"2ADBBFQ5OII5","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_16","alias_value":"2ADBBFQ5OII5JMZB","created_at":"2026-07-05T00:39:32Z"},{"alias_kind":"pith_short_8","alias_value":"2ADBBFQ5","created_at":"2026-07-05T00:39:32Z"}],"graph_snapshots":[{"event_id":"sha256:2c3d7687c1c2468212e7451141dae05cd2ec8a07ab35cd697c52c9220424de14","target":"graph","created_at":"2026-07-05T00:39:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.08732/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated to grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposi","authors_text":"Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-23T09:35:31Z","title":"Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08732","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2b9f4208ba8c792429baaa5b82c4e57440f3ad3074de22374dd87bf73c3dab0a","target":"record","created_at":"2026-07-05T00:39:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"63cbbaa8c46d765c3d5cf442da30ad7ed78c9583182a74f7dd163dd984a22c08","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-23T09:35:31Z","title_canon_sha256":"84e393cdaa0e72c082a235d1d3c4a37341fc3346c6231f288785e28318be36a3"},"schema_version":"1.0","source":{"id":"1908.08732","kind":"arxiv","version":2}},"canonical_sha256":"d00610961d7211d4b32160f5c906863ffffd7343185c5a9c9ce05294dbff7738","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d00610961d7211d4b32160f5c906863ffffd7343185c5a9c9ce05294dbff7738","first_computed_at":"2026-07-05T00:39:32.303247Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:39:32.303247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FKqFyKvXNjsa4Ou8kQ9Ua1FhoOrpZdx9XC9NJO9Ffn+JF2yZkedExItxfhFUtljMR/EKORygrt3PKZ/Wy5jiBw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:39:32.303739Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08732","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b9f4208ba8c792429baaa5b82c4e57440f3ad3074de22374dd87bf73c3dab0a","sha256:2c3d7687c1c2468212e7451141dae05cd2ec8a07ab35cd697c52c9220424de14"],"state_sha256":"52bc55bd09f4e8b6a7e614da5a20bd5480ec6d722bf2a1f5843c8421063fd702"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fnLknRBBDVTZ9mRMnUaQSYlkBAbhARTC2mqh+DTjsVgGRZA6cPc+W/pJjKZQ+mP+KszyKvNcY5IaMx6fDSlBCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T19:00:58.770676Z","bundle_sha256":"e7a8403bc7162b1e4700079ac787070777e2407805e750fe4e6f947030971b2b"}}