{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:06:31Z","timestamp":1760709991145,"version":"3.40.5"},"reference-count":27,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level.\nIn this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones.\nWe shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods.\nWe first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence.\nNumerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy.<\/jats:p>","DOI":"10.1515\/cmam-2018-0188","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T09:59:41Z","timestamp":1566986381000},"page":"397-417","source":"Crossref","is-referenced-by-count":6,"title":["Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6266-373X","authenticated-orcid":false,"given":"Mohammad","family":"Al-Khaleel","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics , Khalifa University , Abu Dhabi , United Arab Emirates ; and Department of Mathematics, Yarmouk University, Irbid, Jordan"}]},{"given":"Shu-Lin","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Science , Sichuan University of Science and Engineering , Zigong , Sichuan 643000 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2019,8,28]]},"reference":[{"key":"2023033110434226887_j_cmam-2018-0188_ref_001","unstructured":"M. J.  Ablowitz and A. S.  Fokas,\nComplex Variables: Introduction and Applications,\nCambridge Texts Appl. Math.,\nCambridge University, Cambridge, 1997."},{"key":"2023033110434226887_j_cmam-2018-0188_ref_002","unstructured":"M. D.  Al-Khaleel,\nOptimized Waveform Relaxation Methods for Circuit Simulations,\nProQuest LLC, Ann Arbor, 2007;\nThesis (Ph.D.)\u2013McGill University (Canada)."},{"key":"2023033110434226887_j_cmam-2018-0188_ref_003","doi-asserted-by":"crossref","unstructured":"T. M.  Atanackovic, S.  Pilipovic and D.  Zorica,\nA diffusion wave equation with two fractional derivatives of different order,\nJ. Phys. A 40 (2007), no. 20, 5319\u20135333.","DOI":"10.1088\/1751-8113\/40\/20\/006"},{"key":"2023033110434226887_j_cmam-2018-0188_ref_004","doi-asserted-by":"crossref","unstructured":"D.  Bennequin, M. J.  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Sci. 60,\nSpringer, Berlin (2006), 203\u2013210.","DOI":"10.1007\/978-3-540-75199-1_22"},{"key":"2023033110434226887_j_cmam-2018-0188_ref_011","doi-asserted-by":"crossref","unstructured":"M. J.  Gander, L.  Halpern and F.  Magoul\u00e8s,\nAn optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation,\nInternat. J. Numer. Methods Fluids 55 (2007), no. 2, 163\u2013175.","DOI":"10.1002\/fld.1433"},{"key":"2023033110434226887_j_cmam-2018-0188_ref_012","unstructured":"M. J.  Gander, L.  Halpern and F.  Nataf,\nOptimal convergence for overlapping and non-overlapping Schwarz waveform relaxation,\nProceedings of the 11th International Conference on Domain Decomposition Methods,\nLecture Notes in Comput. Sci.,\nSpringer, Berlin (1999), 27\u201336."},{"key":"2023033110434226887_j_cmam-2018-0188_ref_013","doi-asserted-by":"crossref","unstructured":"M. J.  Gander, L.  Halpern and F.  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