{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:56:02Z","timestamp":1787381762530,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Phase-based upstreaming, which is a commonly used spatial discretization for multiphase flow in reservoir simulation, naturally gives rise to implicit monotone schemes when implicit time-stepping is used. The nonlinear Gauss\u2013Seidel and Jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations. Thus, for one-dimensional problems, we obtain an alternate, constructive proof that such schemes are well-defined and converge to the entropy solution of the conservation law for arbitrary CFL numbers. The accuracy of phase-based upstream solutions is studied for various spatial and temporal grids, revealing the importance of unconditional stability when nonuniform grids and\/or variable porosity is involved.<\/jats:p>","DOI":"10.1137\/070703922","type":"journal-article","created":{"date-parts":[[2008,7,10]],"date-time":"2008-07-10T14:11:44Z","timestamp":1215699104000},"page":"2662-2687","source":"Crossref","is-referenced-by-count":19,"title":["Convergence of Implicit Monotone Schemes with Applications in Multiphase Flow in Porous Media"],"prefix":"10.1137","volume":"46","author":[{"given":"Felix","family":"Kwok","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamdi A.","family":"Tchelepi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,7,2]]},"reference":[{"key":"R1","unstructured":"H. Abelson, J. Sussman, and J. Sussman,\n                      Structure and Interpretation of Computer Programs\n                      , MIT Press, Cambridge, MA, 1984."},{"key":"R2","unstructured":"K. Aziz and A. Settari,\n                      Petroleum Reservoir Simulation\n                      , Applied Science Publishers, New York, 1979."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0728036"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.2118\/72469-PA"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.2307\/2373376"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"K. Deimling,\n                      Nonlinear Functional Analysis\n                      , Springer-Verlag, Berlin, 1985.","DOI":"10.1007\/978-3-662-00547-7"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"S. Evje and K. H. Karlsen,\n                      Degenerate convection-diffusion equations and implicit monotone difference schemes\n                      , in Hyperbolic Problems: Theory, Numerics, Applications, Internat. Ser. Numer. Math. 129, M. Fey and R. Jeltsch, eds., Birkh\u00e4user Verlag, Basel, 1999, pp. 285\u2013294.","DOI":"10.1007\/978-3-0348-8720-5_31"},{"key":"R8","unstructured":"F. Kwok,\n                      Scalable Linear and Nonlinear Algorithms for Multiphase Flow in Porous Media\n                      , Ph.D. thesis, Stanford University, Stanford, CA, 2007."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842121-6"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0721016"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(70)90298-2"},{"key":"R12","unstructured":"H. L. Royden,\n                      Real Analysis\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1988."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , 2nd ed., SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1983-0679435-6"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070703922","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:40Z","timestamp":1787325280000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070703922"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":14,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070703922"],"URL":"https:\/\/doi.org\/10.1137\/070703922","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}