{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-second-order-formulation-of-the-p_n","title":"The second-order formulation of the $P_N$ equations with Marshak boundary conditions","arxiv_id":"1911.00468","date":"2019-11-01","proceeding":null,"authors":["Matthias Andres","Florian Schneider"],"abstract":"We consider a reformulation of the classical $P_N$ method with Marshak boundary conditions for the approximation of the monoenergetic stationary linear transport equation as a system of second-order PDEs. Our derivation allows the automatic generation of a model hierarchy which can then be handed to standard PDE tools. This method allows for heterogeneous coefficients, irregular grids, anisotropic boundary sources and anisotropic scattering. The wide applicability is demonstrated in several numerical test cases. We make our implementation available online, which allows for fast prototyping.","url_abs":"https://arxiv.org/abs/1911.00468v1","url_pdf":"https://arxiv.org/pdf/1911.00468v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-second-order-formulation-of-the-p_n","repo_url":"https://github.com/andresmatthias/pn2nd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}