{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:37:24Z","timestamp":1787330244868,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>\n                    We consider a linear, Schr\u00f6dinger-type partial differential equation, the \"parabolic\" equation of underwater acoustics, in a layer of water bounded below by a rigid bottom of variable topography. Using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori H\n                    <jats:sup>1<\/jats:sup>\n                    estimate and prove an optimal-order error bound in the maximum norm for a Crank--Nicolson-type finite difference approximation of its solution. We also consider the same problem with an alternative rigid bottom boundary condition due to Abrahamsson and Kreiss and prove again a priori H\n                    <jats:sup>1<\/jats:sup>\n                    estimates and optimal-order error bounds for a Crank--Nicolson scheme.\n                  <\/jats:p>","DOI":"10.1137\/s0036142999367460","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"539-565","source":"Crossref","is-referenced-by-count":10,"title":["Finite Difference Schemes for the \"Parabolic\" Equation in a Variable Depth Environment with a Rigid Bottom Boundary Condition"],"prefix":"10.1137","volume":"39","author":[{"given":"G. D.","family":"Akrivis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V. A.","family":"Dougalis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. E.","family":"Zouraris","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":"10.1002\/mma.1670130503"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1121\/1.399089"},{"key":"R3","first-page":"19","volume":"31","author":"Akrivis Georgios","year":"1990","journal-title":"Bull. Soc. Math. Gr\u00e8ce (N.S.)"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1991-1066829-7"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994266352"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0148005"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218396X96000209"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1121\/1.398829"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1121\/1.400526"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218396X96000222"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0717045"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1121\/1.399448"},{"key":"R13","first-page":"18","volume":"85","author":"Kampanis N. A.","year":"1999","journal-title":"Acustica"},{"key":"R14","unstructured":"I. Karasalo and A. Sundstr\u00f6m,\n                      JEPE \u2010 A high\u2010order PE\u2010model for range\u2010dependent fluid media\n                      , in Proceedings of the Third European Conference on Underwater Acoustics, J. S. Papadakis, ed., Institute of Applied and Computational Mathematics, FO. R. T. H., Heraklion, Greece, 1996, pp. 189\u2013194."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1121\/1.386918"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1121\/1.389431"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(87)90207-0"},{"key":"R18","unstructured":"Mark Robinson, Graeme Fairweather, An orthogonal spline collocation method for the numerical solution of underwater acoustic wave propagation problems, North\u2010Holland, Amsterdam, 1993, 339\u20133531208556"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1121\/1.385072"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1063\/1.525033"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1121\/1.400525"},{"key":"R22","unstructured":"F. Sturm,\n                      Mod\u00e9lisation math\u00e9matique et num\u00e9rique d\u2019un probl\u00e8me de propagation en acoustique sous\u2010marine: prise en compte d\u2019un environnement variable tridimensionnel\n                      , Th\u00e8se de Docteur en Sciences Universit\u00e9 de Toulon et du Var, France, 1997."},{"key":"R23","unstructured":"F. B. Sturm, J. A. Fawcett, and M.\u2010C. P\u00e9lissier,\n                      Application to 3D parabolic models to benchmark problems\n                      , in Proceedings of the Fourth European Conference on Underwater Acoustics, A. A. Alippi and G. B. Canelli, eds., CNR\u2010IDAC, Rome, 1998, pp. 733\u2013738."},{"key":"R24","unstructured":"F. Sturm, M. C. Pelissier and D. Fattaccioli,\n                      3\u2010D sound propagation modelling with TRIPARADIM\n                      , Proceedings of the Third European Conference on Underwater Acoustics J. S. Papadakis, ed., Institute of Applied and Computational Mathematics, FO. R. T. H., Heraklion, Greece, 1996, pp. 244\u2013248."},{"key":"R25","doi-asserted-by":"crossref","unstructured":"Fred Tappert, The parabolic approximation method, Springer, Berlin, 1977, 0\u20130, 224\u2013287. Lecture Notes in Phys., Vol. 7057:14891","DOI":"10.1007\/3-540-08527-0_5"},{"key":"R26","unstructured":"G. E. Zouraris,\n                      A Conservative Crank\u2013Nicolson\u2013Type Finite Difference Scheme for the Linear Schr\u00f6dinger Equation in a Noncylindrical Domain\n                      , Technical Report TR1TA\u2010NA\u20109808, Department of Numerical Analysis and Computing Science (NADA), Royal Institute of Technology (KTH), Stockholm, Sweden, 1998."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0036142999367460","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:53:57Z","timestamp":1787327637000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0036142999367460"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,1]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["10.1137\/S0036142999367460"],"URL":"https:\/\/doi.org\/10.1137\/s0036142999367460","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,1]]}}}