{"paper":{"title":"An Energy-Stable Implicit Convex-Splitting BDF2 Scheme for the Cahn-Hilliard-Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Qi Wang, Xuelong Gu","submitted_at":"2026-06-02T20:49:14Z","abstract_excerpt":"We develop an energy-stable implicit convex-splitting BDF2 discretization (CS-BDF2) of the Cahn--Hilliard--Navier--Stokes equations. For the Cahn--Hilliard equation, BDF2 analyses can establish energy stability by testing the phase equation in the (H^{-1}) metric. For CHNS, this test is not compatible with the coupled energy estimate: the momentum equation is tested by (\\bfu^{n+1}), while the transported phase equation is tested by (\\mu^{n+1}) so that transport cancels capillary work. The chemical-potential relation must then be paired with the BDF2 phase increment ((3\\phi^{n+1}-4\\phi^n+\\phi^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04204","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/2606.04204/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"}