{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:09:39Z","timestamp":1759838979787,"version":"3.41.2"},"reference-count":25,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,12,28]],"date-time":"2011-12-28T00:00:00Z","timestamp":1325030400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11026040"],"award-info":[{"award-number":["11026040"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>The finite difference method discretization of Helmholtz equations usually leads to\nthe large spare linear systems. Since the coefficient matrix is frequently indefinite, it\nis difficult to solve iteratively. In this paper, a modified symmetric successive overrelaxation (MSSOR) preconditioning strategy is constructed based on the coefficient\nmatrix and employed to speed up the convergence rate of iterative methods. The idea\nis to increase the values of diagonal elements of the coefficient matrix to obtain better\npreconditioners for the original linear systems. Compared with SSOR preconditioner,\nMSSOR preconditioner has no additional computational cost to improve the convergence rate of iterative methods. Numerical results demonstrate that this method can\nreduce both the number of iterations and the computational time significantly with\nlow cost for construction and implementation of preconditioners.<\/jats:p>","DOI":"10.1155\/2012\/365124","type":"journal-article","created":{"date-parts":[[2011,12,28]],"date-time":"2011-12-28T21:05:09Z","timestamp":1325106309000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Modified SSOR Preconditioning Strategy for Helmholtz Equations"],"prefix":"10.1155","volume":"2012","author":[{"given":"Shi-Liang","family":"Wu","sequence":"first","affiliation":[]},{"given":"Cui-Xia","family":"Li","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2011,12,28]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(95)00094-1"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2004.01.009"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(83)90139-0"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503428539"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/nla.615"},{"volume-title":"Iterative Methods for Sparse Linear Systems","year":"1996","author":"Saad Y.","key":"e_1_2_6_6_2"},{"key":"e_1_2_6_7_2","first-page":"425","volume-title":"Advances in Computer Methods for Partial Differential Equations","author":"Gozani J.","year":"1984"},{"key":"e_1_2_6_8_2","unstructured":"LairdA. 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