{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:56Z","timestamp":1787337296733,"version":"3.56.0"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.<\/jats:p>","DOI":"10.1137\/070681867","type":"journal-article","created":{"date-parts":[[2008,2,4]],"date-time":"2008-02-04T11:37:29Z","timestamp":1202125049000},"page":"463-472","source":"Crossref","is-referenced-by-count":95,"title":["Iterative Solution of Piecewise Linear Systems"],"prefix":"10.1137","volume":"30","author":[{"given":"Luigi","family":"Brugnano","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vincenzo","family":"Casulli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,1]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"J. Bear and A. Verruijt,\n                      Modeling Groundwater Flow and Pollution\n                      , D. Reidel, Dordrecht, Holland, 1987.","DOI":"10.1007\/978-94-009-3379-8"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00432-5"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90091-E"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(02)00264-9"},{"key":"R5","unstructured":"R. W. Cottle, J.S. Pang, and R. E. Stone,\n                      The Linear Complementarity Problem\n                      , Academic Press, San Diego, 1992."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.17.9.612"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02592192"},{"key":"R8","unstructured":"G. H. Golub and C. F. van Loan,\n                      Matrix Computations\n                      , 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson,\n                      Topics in Matrix Analysis\n                      , Cambridge University Press, New York, 1991.","DOI":"10.1017\/CBO9780511840371"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1080\/10556780310001636369"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"C. E. Lemke and J. T. Howson, Jr.\n                      Equilibrium points of bimatrix games\n                      , J. Soc. Indust. Appl. Math. 12 (1964), pp. 413\u2013423.","DOI":"10.1137\/0112033"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/050623723"},{"key":"R13","unstructured":"K. G. Murty,\n                      Linear Complementarity, Linear and Nonlinear Programming\n                      , Heldermann Verlag, Berlin, 1988 (http:\/\/ioe.engin.umich.edu\/people\/fac\/books\/murty\/linear_complementarity_webbook\/)."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-003-0472-9"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00433-7"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1287\/moor.18.1.227"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581275"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems, Second Edition\n                      , SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1002\/fld.537"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"S. J. Wright,\n                      Primal-Dual Interior-Point Methods\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971453"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070681867","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:55:01Z","timestamp":1787334901000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070681867"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070681867"],"URL":"https:\/\/doi.org\/10.1137\/070681867","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}