{"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/on-optimal-h-independent-convergence-of","title":"On \"Optimal\" h-Independent Convergence of Parareal and MGRIT using Runge-Kutta Time Integration","arxiv_id":"1906.06672","date":"2019-06-16","proceeding":null,"authors":["Stephanie Friedhoff","Ben S. Southworth"],"abstract":"Although convergence of the Parareal and multigrid-reduction-in-time (MGRIT) parallel-in-time algorithms is well studied, results on their optimality is limited. Appealling to recently derived tight bounds of two-level Parareal and MGRIT convergence, this paper proves (or disproves) $h_x$- and $h_t$-independent convergence of two-level Parareal and MGRIT, for linear problems of the form $\\mathbf{u}'(t) + \\mathcal{L}\\mathbf{u}(t) = f(t)$, where $\\mathcal{L}$ is symmetric positive definite and Runge-Kutta time integration is used. The theory presented in this paper also encompasses analysis of some modified Parareal algorithms, such as the $\\theta$-Parareal method, and shows that not all Runge-Kutta schemes are equal from the perspective of parallel-in-time. Some schemes, particularly L-stable methods, offer significantly better convergence than others as they are guaranteed to converge rapidly at both limits of small and large $h_t\\xi$, where $\\xi$ denotes an eigenvalue of $\\mathcal{L}$ and $h_t$ time-step size. On the other hand, some schemes do not obtain $h$-optimal convergence, and two-level convergence is restricted to certain regimes. In certain cases, an $\\mathcal{O}(1)$ factor change in time step $h_t$ or coarsening factor $k$ can be the difference between convergence factors $\\rho\\approx0.02$ and divergence! The analysis is extended to skew symmetric operators as well, which cannot obtain $h$-independent convergence and, in fact, will generally not converge for a sufficiently large number of time steps. Numerical results confirm the analysis in practice and emphasize the importance of a priori analysis in choosing an effective coarse-grid scheme and coarsening factor. A Mathematica notebook to perform a priori two-grid analysis is available at https://github.com/XBraid/xbraid-convergence-est.","url_abs":"http://arxiv.org/abs/1906.06672v2","url_pdf":"http://arxiv.org/pdf/1906.06672v2.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":"on-optimal-h-independent-convergence-of","repo_url":"https://github.com/XBraid/xbraid-convergence-est","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}