{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:42:52Z","timestamp":1776789772539,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,7,1]],"date-time":"2009-07-01T00:00:00Z","timestamp":1246406400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for semilinear wave propagation in one dimension. Using the theory of absorbing boundary conditions, we derive a new nonlinear algorithm. We show that the algorithm is well-posed and we prove its convergence by energy estimates and a Galerkin method. We then introduce an explicit scheme. We prove the convergence of the discrete algorithm with suitable assumptions on the nonlinearity. We finally illustrate our analysis with numerical experiments.<\/p>","DOI":"10.1090\/s0025-5718-08-02164-9","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"865-889","source":"Crossref","is-referenced-by-count":14,"title":["Nonlinear nonoverlapping Schwarz waveform relaxation for semilinear wave propagation"],"prefix":"10.1090","volume":"78","author":[{"given":"Laurence","family":"Halpern","sequence":"first","affiliation":[]},{"given":"J\u00e9r\u00e9mie","family":"Szeftel","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,7,1]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"crossref","first-page":"209","DOI":"10.24033\/asens.1404","article-title":"Calcul symbolique et propagation des singularit\u00e9s pour les \u00e9quations aux d\u00e9riv\u00e9es partielles non lin\u00e9aires","volume":"14","author":"Bony, Jean-Michel","year":"1981","journal-title":"Ann. 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J.","year":"1999"},{"issue":"5","key":"7","doi-asserted-by":"publisher","first-page":"1643","DOI":"10.1137\/S003614290139559X","article-title":"Optimal Schwarz waveform relaxation for the one dimensional wave equation","volume":"41","author":"Gander, Martin J.","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"8","isbn-type":"print","first-page":"1","article-title":"On the Schwarz alternating method. I","author":"Lions, P.-L.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712203"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"39","DOI":"10.5802\/aif.1037","article-title":"R\u00e9gularit\u00e9 microlocale pour des probl\u00e8mes aux limites non lin\u00e9aires","volume":"36","author":"Sabl\u00e9-Tougeron, Monique","year":"1986","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"10","unstructured":"H. A. Schwarz, \u00dcber einen Grenz\u00fcbergang durch alternierendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Z\u00fcrich 15 (1870), 272\u2013286."},{"issue":"29-32","key":"11","doi-asserted-by":"publisher","first-page":"3760","DOI":"10.1016\/j.cma.2005.03.009","article-title":"Absorbing boundary conditions for nonlinear scalar partial differential equations","volume":"195","author":"Szeftel, J\u00e9r\u00e9mie","year":"2006","journal-title":"Comput. Methods Appl. Mech. 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