{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:23Z","timestamp":1787336903488,"version":"build-2736575974"},"reference-count":26,"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":[[2000,1]]},"abstract":"<jats:p>This work represents a step toward advancing classical methods of bifurcation analysis in conjunction with very large-scale scientific computing needed to model realistically processes which involve laminar and steady flows of incompressible fluids. A robust method of analysis based on pseudoarclength continuation, Newton's method, and direct solution of linear systems is proposed and applied to the analysis of a representative system of fluid mechanics, namely the tilted lid driven cavity. Accurate solution curves, possessing simple singular points, were computed for Reynolds numbers varying from 0 to 10,000 and for different angles of tilt. The results demonstrate that two-dimensional models with up to 1,000,000 algebraic equations can be studied feasibly using the methods described here with state-of-the-art vector supercomputers.<\/jats:p>","DOI":"10.1137\/s1064827598334514","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"285-311","source":"Crossref","is-referenced-by-count":11,"title":["Construction of Solution Curves for Large Two-Dimensional Problems of Steady-State Flows of Incompressible Fluids"],"prefix":"10.1137","volume":"22","author":[{"given":"Valmor F.","family":"De Almeida","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeffrey J.","family":"Derby","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61257-2"},{"key":"R2","unstructured":"V. F. de Almeida and J. J. Derby (1997),\n                      Construction of Solution Curves for Large 2\u2010D\n                      Problems of Steady\u2010State Flows of Incompressible Fluids, Minnesota Supercomputer Institute Research Report 97\/070, University of Minnesota, Department of Chemical Engineering and Materials Science, Minneapolis, MN 55455\u20100132, 1997;"},{"key":"R2","unstructured":"also available at (de_almeida@na\u2010net.ornl.gov) by request or at http:\/\/www.msi.umn.edu\/\u02dcdalmeida."},{"key":"R3","unstructured":"shape V. F. de Almeida and J. J. Derby (1999),\n                      Preferred Method for Setting a Pressure Level in Computations of 3\u2010D Flows of Incompressible Fluids with GFEM\n                      , Minnesota Supercomputer Institute Research Report 99\/005, University of Minnesota, Department of Chemical Engineering and Materials Science, Minneapolis, MN; also available at (de_almeida@na\u2010net.ornl.gov) by request and at http:\/\/www.msi.umn.edu\/\u02dcdalmeida."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90148-9"},{"key":"R5","unstructured":"shape R. Brent (1973),\n                      Some efficient algorithms for solving systems of nonlinear equations\n                      , SIAM J. Numer. Anal., 10, pp. 327\u201344."},{"key":"R6","unstructured":"R. Hoppe, B. Wohlmuth, Adaptive mixed hybrid and macro\u2010hybrid finite element methods, Proceedings of the Algoritmy\u201897 Conference on Scientific Computing (Zuberec), Vol. 67, 1998, 159\u20131792000a:65161"},{"key":"R7","unstructured":"Tony Chan, Tarek Mathew, Domain decomposition algorithms, Acta Numer., Cambridge Univ. Press, Cambridge, 1994, 61\u201314395f:65214"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0724059"},{"key":"R9","unstructured":"shape I. S. Duff (1996),\n                      Sparse Numerical Algebra: Direct Methods and Preconditioning\n                      , Research Report RAL\u2010TR 96\u2010047, Department for Computation and Information, Atlas Centre, Rutherford Appleton Laboratory, Oxon, U. K."},{"key":"R10","unstructured":"shape I. S. Duff and J. A. Scott (1993),\n                      MA42\u2014A New Frontal Code for Solving Sparse Unsymmetric Systems\n                      , Research Report RAL 93\u2010064, Central Computing Department, Atlas Centre, Rutherford Appleton Laboratory, Oxon, U. K."},{"key":"R11","unstructured":"shape I. S. Duff and J. A. Scott (1994),\n                      The Use of Multiple Fronts in Gaussian Elimination\n                      , Research Report RAL 94\u2010040, Central Computing Department, Rutherford Appleton Laboratory, Oxon, U. K."},{"key":"R12","unstructured":"shape I. S. Duff, J. K. Reid, and J. A. Scott (1989),\n                      The use of profile algorithms with a frontal code\n                      , Internat. J. Numer. Methods Engrg., 28, pp. 2555\u20132568."},{"key":"R13","unstructured":"shape A. Fortin, M. Fortin, and J. Gervais (1987),\n                      A numerical simulation of the transition to turbulence in a two\u2010dimensional flow\n                      , J. Comp. Phys., 70, pp. 295\u2013310."},{"key":"R14","unstructured":"shape M. Fortin and A. Fortin (1985),\n                      A generalization of Uzawa\u2019s algorithm for the solution of the Navier\u2010Stokes equations\n                      , Commun. Appl. Numer. Methods, 1, pp. 205\u2013208."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0913025"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(91)90015-V"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0718066"},{"key":"R18","unstructured":"shape G. Karypis (1996),\n                      Graph Partitioning and Its Applications to Scientific Computing\n                      , Ph.D. dissertation, University of Minnesota, Department of Computer Science."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595287997"},{"key":"R20","volume-title":"Numerical methods for two\u2010point boundary value problems","author":"Keller Herbert","year":"1992"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0717020"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(94)90029-9"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/0724089"},{"key":"R24","unstructured":"shape V. E. Shamanskii (1967),\n                      A modification of Newton\u2019s method\n                      , Ukran. Mat. Zh., 19, pp. 133\u2013138 (in Russian)."},{"key":"R25","unstructured":"shape S. W. Sloan (1989),\n                      A fortran program for profile and wavefront reduction\n                      , Internat. J. Numer. Methods Engrg., 28, pp. 2651\u20132679."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827598334514","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:56Z","timestamp":1787333756000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827598334514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827598334514"],"URL":"https:\/\/doi.org\/10.1137\/s1064827598334514","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}