{"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/semi-implicit-explicit-runge-kutta-method-for","title":"Semi-implicit-explicit Runge-Kutta method for nonlinear differential equations","arxiv_id":"2504.09969","date":"2025-04-14","proceeding":null,"authors":["Lingyun Ding"],"abstract":"A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\\mathbf{u}' = \\mathbf{f}(t,\\mathbf{u}) + G(t,\\mathbf{u}) \\mathbf{u}$, where $\\mathbf{f}$ is a non-stiff term and $G\\mathbf{u}$ represents the stiff terms. Such systems frequently arise from spatial discretizations of time-dependent nonlinear partial differential equations (PDEs). For instance, $G$ could involve higher-order derivative terms with nonlinear coefficients. Traditional IMEX-RK methods, which treat $\\mathbf{f}$ explicitly and $G\\mathbf{u}$ implicitly, require solving nonlinear systems at each time step when $G$ depends on $\\mathbf{u}$, leading to increased computational cost and complexity. In contrast, the proposed semi-IMEX scheme treats $G$ explicitly while keeping $\\mathbf{u}$ implicit, reducing the problem to solving only linear systems. This approach eliminates the need to compute Jacobians while preserving the stability advantages of implicit methods. A family of semi-IMEX RK schemes with varying orders of accuracy is introduced. Numerical simulations for various nonlinear equations, including nonlinear diffusion models, the Navier-Stokes equations, and the Cahn-Hilliard equation, confirm the expected convergence rates and demonstrate that the proposed method allows for larger time step sizes without triggering stability issues.","url_abs":"https://arxiv.org/abs/2504.09969v1","url_pdf":"https://arxiv.org/pdf/2504.09969v1.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":"semi-implicit-explicit-runge-kutta-method-for","repo_url":"https://github.com/wdachub/semi-imex-rk-method","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}