{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T01:41:14Z","timestamp":1790473274344,"version":"4.1.0"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["DE-AC52-07NA27344"],"award-info":[{"award-number":["DE-AC52-07NA27344"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["LLNL-JRNL-692418"],"award-info":[{"award-number":["LLNL-JRNL-692418"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [ XBraid: Parallel Multigrid in Time, http:\/\/llnl.gov\/casc\/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge, and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. Second, we explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the diffusion dominated problems considered here.<\/jats:p>","DOI":"10.1137\/16m1074096","type":"journal-article","created":{"date-parts":[[2017,10,26]],"date-time":"2017-10-26T16:24:40Z","timestamp":1509035080000},"page":"S501-S527","source":"Crossref","is-referenced-by-count":69,"title":["Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT)"],"prefix":"10.1137","volume":"39","author":[{"given":"V. A.","family":"Dobrev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tz.","family":"Kolev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N. A.","family":"Petersson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. B.","family":"Schroder","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,10,26]]},"reference":[{"key":"atypb1","unstructured":"G. Bal,\n                      On the convergence and the stability of the parareal algorithm to solve partial differential equations\n                      , in Domain Decomposition Methods in Science and Engineering, Lecture Notes in Comput. Sci. Eng. 40, T. J. Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose, T. Schlick, R. Kornhuber, R. Hoppe, J. P\u00e9riaux, O. Pironneau, O. Widlund, and J. Xu, eds., Springer, Berlin, Heidelberg, 2005, pp. 425-432."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.2514\/6.1979-1455"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719505"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/110861002"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503430758"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.2140\/camcos.2012.7.105"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/130944230"},{"key":"atypb8","volume-title":"Lawrence Livermore National Laboratory","author":"Falgout R. 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Magoul\u00e8s, ed., Saxe-Coburg Publications, Stirlingshire, UK"},{"key":"atypb17","unstructured":"MFEM: Modular Finite Element Methods\n                      , version 3.2,http:\/\/mfem.org, 2017."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.2140\/camcos.2010.5.265"},{"key":"atypb19","first-page":"388","volume-title":"AIP Conf. Proc. 1048","author":"Minion M. L.","year":"2008"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1145\/355588.365137"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(83)90091-5"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/SC.2012.6"},{"key":"atypb23","unstructured":"G. A. Staff and E. M. R\u00f8 nquist,\n                      Stability of the parareal algorithm\n                      , in Domain Decomposition Methods in Science and Engineering, Lecture Notes in Comput. Sci. Eng. 40, T. J. Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose, T. Schlick, R. Kornhuber, R. Hoppe, J. P\u00e9riaux, O. Pironneau, O. Widlund, and J. Xu, eds., Springer, Berlin, Heidelberg, 2005, pp. 449-456."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238230"},{"key":"atypb25","unstructured":"XBraid: Parallel Multigrid in Time\n                      , version 2.1.0,http:\/\/llnl.gov\/casc\/xbraid, 2016."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596302989"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1074096","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:10:19Z","timestamp":1787328619000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1074096"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":26,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1074096"],"URL":"https:\/\/doi.org\/10.1137\/16m1074096","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}