{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:32:58Z","timestamp":1787329978886,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2015M580777"],"award-info":[{"award-number":["2015M580777"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2016T90841"],"award-info":[{"award-number":["2016T90841"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004829","name":"Department of Science and Technology of Sichuan Province","doi-asserted-by":"publisher","award":["2014JQ0035"],"award-info":[{"award-number":["2014JQ0035"]}],"id":[{"id":"10.13039\/501100004829","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004829","name":"Department of Science and Technology of Sichuan Province","doi-asserted-by":"publisher","award":["15ZA0220"],"award-info":[{"award-number":["15ZA0220"]}],"id":[{"id":"10.13039\/501100004829","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11301362"],"award-info":[{"award-number":["11301362"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61573010"],"award-info":[{"award-number":["61573010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671074"],"award-info":[{"award-number":["11671074"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004858","name":"Sichuan University of Science and Engineering","doi-asserted-by":"publisher","award":["2015LX01"],"award-info":[{"award-number":["2015LX01"]}],"id":[{"id":"10.13039\/501100004858","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>It is well known that the optimal transmission conditions (TCs) give (in theory) the best convergence for the Schwarz waveform relaxation (SWR) algorithm. These TCs are of convolution type and are not systematically explored yet, partly because numerical computation of convolutions is expensive. However, if the governing equation is also of convolution type, such as the Volterra integro-partial differential equations with weakly singular kernels studied in this paper, this kind of TCs seems a very natural choice. For a class of convolution TCs including the optimal one derived from the space-time continuous analysis, we perform a convergence analysis for the SWR algorithm at the fully discretized level in the framework of convolution quadratures, which permits us to precisely capture the effects of the discretizations on the convergence rate. By optimizing the parameter in the TCs, we show that the convergence factor can be reduced to a value around 0.0730 if an A-stable convolution quadrature is used. This result is independent of the mesh parameters $\\Delta t$ and $\\Delta x$ and is the worst estimate of the convergence factor. In some situations, for example, the case that an L-stable convolution quadrature is used and $\\Delta t=O(\\Delta x)$, the convergence factor tends to zero as $\\Delta x\\rightarrow0$. Numerical results are given to support our theoretical analysis.<\/jats:p>","DOI":"10.1137\/16m1072620","type":"journal-article","created":{"date-parts":[[2017,5,23]],"date-time":"2017-05-23T11:55:26Z","timestamp":1495540526000},"page":"A890-A921","source":"Crossref","is-referenced-by-count":8,"title":["Convergence Analysis of Schwarz Waveform Relaxation with Convolution Transmission Conditions"],"prefix":"10.1137","volume":"39","author":[{"given":"Shu-Lin","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingxiang","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,5,23]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2003.v1.n3.a7"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/110854187"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2008.2008286"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543234"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/110842995"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0784-8"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-08-02145-5"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-06-01788-1"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2014.09.024"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596305337"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100345"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/050642137"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2004.826193"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01398686"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01462237"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1023\/B:BITN.0000046813.23911.2d"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501388741"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/080736533"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/TCAD.1982.1270004"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/070691450"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2004.08.022"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/S0370-1573(00)00070-3"},{"key":"atypb23","volume-title":"Fractional Differential Equations","author":"Podlubny I.","year":"1999"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/050623139"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/b137868"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-26825-1_28"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-014-0475-3"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/110860975"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"S. L. Wu,\n                      Overlapping Schwarz waveform relaxation for a class of time-fractional diffusion problems\n                      , J. Sci. Comput., to appear,https:\/\/doi.org\/10.1007\/s10915-017-0379-x.","DOI":"10.1007\/s10915-017-0379-x"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1072620","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:42:02Z","timestamp":1787326922000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1072620"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1072620"],"URL":"https:\/\/doi.org\/10.1137\/16m1072620","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}