{"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/high-order-numerical-integration-on-domains","title":"High-Order Numerical Integration on Domains Bounded by Intersecting Level Sets","arxiv_id":"2308.10698","date":"2023-08-21","proceeding":null,"authors":["Lauritz Beck","Florian Kummer"],"abstract":"We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on hypercubes to the curved domains of the integrals. This enables the numerical integration of a wide range of integrands since integration on hypercubes is a well known problem. The mappings are constructed by treating the isocontours of the level sets as graphs of height functions. Numerical experiments with smooth integrands indicate a high-order of convergence for transformed Gauss quadrature rules on domains defined by polynomial, rational, and trigonometric level sets. We show that the approach we have used can be combined readily with adaptive quadrature methods. Moreover, we apply the approach to numerically integrate on difficult geometries without requiring a low-order fallback method.","url_abs":"https://arxiv.org/abs/2308.10698v1","url_pdf":"https://arxiv.org/pdf/2308.10698v1.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":"high-order-numerical-integration-on-domains","repo_url":"https://github.com/rumbecken/intersectingquadrature","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}