{"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/accurate-solution-of-the-nonlinear","title":"Accurate Solution of the Nonlinear Schrödinger Equation via Conservative Multiple-Relaxation ImEx Methods","arxiv_id":"2309.02324","date":"2023-09-05","proceeding":null,"authors":["Abhijit Biswas","David I. Ketcheson"],"abstract":"The nonlinear Schr\\\"{o}dinger (NLS) equation possesses an infinite hierarchy of conserved densities and the numerical preservation of some of these quantities is critical for accurate long-time simulations, particularly for multi-soliton solutions. We propose an essentially explicit discretization that conserves one or two of these conserved quantities by combining higher-order Implicit-Explicit (ImEx) Runge-Kutta time integrators with the relaxation technique and adaptive step size control. We show through numerical tests that our mass-conserving method is much more efficient and accurate than the widely-used 2nd-order time-splitting pseudospectral approach. Compared to higher-order operator splitting, it gives similar results in general and significantly better results near the semi-classical limit. Furthermore, for some problems adaptive time stepping provides a dramatic reduction in cost without sacrificing accuracy. We also propose a full discretization that conserves both mass and energy by using a conservative finite element spatial discretization and multiple relaxation in time. Our results suggest that this method provides a qualitative improvement in long-time error growth for multi-soliton solutions.","url_abs":"https://arxiv.org/abs/2309.02324v1","url_pdf":"https://arxiv.org/pdf/2309.02324v1.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":"accurate-solution-of-the-nonlinear","repo_url":"https://github.com/abhibsws/multiple_relaxation_nls","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}