{"paper":{"title":"Implicit Dynamical Tensor Train Approximation for Kinetic Equations with Stiff Fokker--Planck Collisions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An implicit dynamical low-rank method in tensor-train format overcomes stability constraints for kinetic equations with stiff Fokker-Planck collisions.","cross_cats":["cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Geshuo Wang, Jingwei Hu","submitted_at":"2026-05-14T20:12:21Z","abstract_excerpt":"Low-rank methods for kinetic equations have attracted increasing attention due to their effectiveness in reducing the high dimensionality of phase space. In our previous work [G. Wang & J. Hu, J. Comput. Phys. 558 (2026) 114884], we developed a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format, in which explicit time integration is employed in all substeps. As a result, the method is subject to severe stability constraints in the strongly collisional regimes. In this paper, we consider kinetic equations with the (nonlinear) Fokker--Planck collisi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The proposed implicit or IMEX dynamical low-rank method in tensor-train format overcomes the severe stability constraints of explicit schemes in strongly collisional regimes while preserving linear scaling with respect to the number of grid points in a single velocity dimension.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Sylvester equations arising in the implicit substeps admit stable, structure-preserving direct solutions whose accuracy and low-rank property are not degraded by the implicit treatment or by the specific form of the nonlinear Fokker-Planck operator.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An implicit dynamical tensor-train method for stiff kinetic equations with Fokker-Planck collisions that formulates substeps as Sylvester equations solved by direct structured solvers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An implicit dynamical low-rank method in tensor-train format overcomes stability constraints for kinetic equations with stiff Fokker-Planck collisions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"143030dca8763d22a78e8bcf0af199d9189ec10e79033ac552d569e1861de34a"},"source":{"id":"2605.15382","kind":"arxiv","version":1},"verdict":{"id":"d376e782-4909-424d-8c30-0e7f9415bed1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T15:50:18.191678Z","strongest_claim":"The proposed implicit or IMEX dynamical low-rank method in tensor-train format overcomes the severe stability constraints of explicit schemes in strongly collisional regimes while preserving linear scaling with respect to the number of grid points in a single velocity dimension.","one_line_summary":"An implicit dynamical tensor-train method for stiff kinetic equations with Fokker-Planck collisions that formulates substeps as Sylvester equations solved by direct structured solvers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Sylvester equations arising in the implicit substeps admit stable, structure-preserving direct solutions whose accuracy and low-rank property are not degraded by the implicit treatment or by the specific form of the nonlinear Fokker-Planck operator.","pith_extraction_headline":"An implicit dynamical low-rank method in tensor-train format overcomes stability constraints for kinetic equations with stiff Fokker-Planck collisions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15382/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T16:04:40.982984Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T16:01:18.048488Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.177335Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.729039Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"761660139815edfa0d1840e1a6de58e8f3aa540d35cc26d2003f24af57fc361e"},"references":{"count":44,"sample":[{"doi":"","year":2025,"title":"Appel¨ o and Y","work_id":"3db6ad1a-5cba-4dae-b5d7-5c230c122dc9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"U. Banik and A. Bhattacharjee. Relaxation of weakly collisional plasma: Continuous spectra, discrete eigenmodes, and the decay of echoes.Phys. Rev. E, 110(4):045204, 2024","work_id":"58bee931-ba54-437c-8419-4b613255f101","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1972,"title":"R. Bartels and G. Stewart. Algorithm 432: Solution of the matrix equationAX+XB=C.Comm. ACM, 15:820–826, 1972","work_id":"b609f701-e66c-4826-b0eb-137864df5792","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"C. Black, K. Germaschewski, A. Bhattacharjee, and C. Ng. Discrete kinetic eigenmode spectra of electron plasma oscillations in weakly collisional plasma: A numerical study.Phys. 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