{"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/isomeric-trees-and-the-order-of-runge-kutta","title":"Isomeric trees and the order of Runge--Kutta methods","arxiv_id":"2112.01608","date":"2021-12-02","proceeding":null,"authors":["John C. Butcher","Helmut Podhaisky"],"abstract":"The conditions for a Runge--Kutta method to be of order $p$ with $p\\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into \"atomic stumps\" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of \"ambiguous orders\" five and six are constructed. These are used to verify the validity of the results.","url_abs":"https://arxiv.org/abs/2112.01608v1","url_pdf":"https://arxiv.org/pdf/2112.01608v1.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":"isomeric-trees-and-the-order-of-runge-kutta","repo_url":"https://github.com/computational-b-series/isomers","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}