{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:04Z","timestamp":1760060104320,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Innovation and Technology NRDI Office","award":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"],"award-info":[{"award-number":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"]}]},{"name":"Thematic Excellence Program","award":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"],"award-info":[{"award-number":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"]}]},{"name":"Digital Government Development and Project Management Ltd.","award":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"],"award-info":[{"award-number":["RRF-2.3.1-21-2022-00004","ELTE TKP 2021-NKTA-62"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we investigate how adaptive time-integration strategies can be effectively combined with parallel-in-time numerical methods for solving systems of ordinary differential equations. Our focus is particularly on their influence on tolerance proportionality. We examine various grid-refinement strategies within the multigrid reduction-in-time (MGRIT) framework. Our results show that a simple adjustment to the original refinement factor can substantially improve computational stability and reliability. Through numerical experiments on standard test problems using the XBraid library, we demonstrate that parallel-in-time solutions closely match their sequential counterparts. Moreover, with the use of multiple processors, computing time can be significantly reduced.<\/jats:p>","DOI":"10.3390\/a18080484","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T07:49:58Z","timestamp":1754380198000},"page":"484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge\u2013Kutta Methods"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8450-7631","authenticated-orcid":false,"given":"Imre","family":"Fekete","sequence":"first","affiliation":[{"name":"Department of Applied Analysis and Computational Mathematics, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P. s. 1\/C, 1117 Budapest, Hungary"},{"name":"Department of Network and Data Science, Central European University, Quellenstra\u00dfe 51, 1100 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7398-7318","authenticated-orcid":false,"given":"Ferenc","family":"Izs\u00e1k","sequence":"additional","affiliation":[{"name":"Department of Applied Analysis and Computational Mathematics, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P. s. 1\/C, 1117 Budapest, Hungary"},{"name":"NumNet HUN-REN\u2013ELTE Research Group, E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P. s. 1\/C, 1117 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vendel P.","family":"Kup\u00e1s","sequence":"additional","affiliation":[{"name":"Department of Applied Analysis and Computational Mathematics, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, P\u00e1zm\u00e1ny P. s. 1\/C, 1117 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3559-960X","authenticated-orcid":false,"given":"Gustaf","family":"S\u00f6derlind","sequence":"additional","affiliation":[{"name":"Department of Numerical Analysis, Center for Mathematical Sciences, Lund University, P.O. 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