{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:37:17Z","timestamp":1787330237433,"version":"build-2736575974"},"reference-count":20,"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":[[2006,1]]},"abstract":"<jats:p>This paper is devoted to the analysis of a new compact scheme for the Navier\u2013Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. Ben\u2010Artzi et al., J. Comput. Phys., 205 (2005), pp. 640\u2013664]. The scheme discussed here combines the Stephenson scheme for the biharmonic operator and ideas from box\u2010scheme methodology. Consistency and convergence are proved for the full nonlinear system. Instead of customary periodic conditions, the case of boundary conditions is addressed. It is shown that in one dimension the truncation error for the biharmonic operator is $O(h^4)$ at interior points and $O(h)$ at near\u2010boundary points. In two dimensions the truncation error is $O(h^2)$ at interior points (due to the cross\u2010terms) and $O(h)$ at near\u2010 boundary points. Hence the scheme is globally of order four in the one\u2010dimensional periodic case and of order two in the two\u2010dimensional periodic case, but of order 3\/2 for one\u2010 and two\u2010 dimensional nonperiodic boundary conditions. We emphasize in particular that there is no special treatment of the boundary, thus allowing robust use of the scheme. The finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications.<\/jats:p>","DOI":"10.1137\/05062915x","type":"journal-article","created":{"date-parts":[[2006,11,16]],"date-time":"2006-11-16T18:00:25Z","timestamp":1163700025000},"page":"1997-2024","source":"Crossref","is-referenced-by-count":29,"title":["Convergence of a Compact Scheme for the Pure Streamfunction Formulation of the Unsteady Navier\u2013Stokes System"],"prefix":"10.1137","volume":"44","author":[{"given":"Matania","family":"Ben\u2010Artzi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean\u2010Pierre","family":"Croisille","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dalia","family":"Fishelov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,10,16]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S1464827596296970"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2426(199707)13:4<375::AID-NUM5>3.0.CO;2-I"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.11.024"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2001117"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6970"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"L. Collatz,\n                      The Numerical Treatment of Differential Equations\n                      , 3rd ed., Springer\u2010Verlag, Berlin, 1960.","DOI":"10.1007\/978-3-642-88434-4"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s006070200002"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(91)90003-O"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.0125"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.fl.23.010191.002213"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.01.002"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1650040704"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0729040"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"H. B. Keller,\n                      A new difference scheme for parabolic problems\n                      , in Numerical Solutions of Partial Differential Equations, II, Academic Press, New York, 1971, pp. 327\u2013350.","DOI":"10.1016\/B978-0-12-358502-8.50014-1"},{"key":"R15","unstructured":"H. Lamb,\n                      Hydrodynamics\n                      , 6th ed., Cambridge University Press, Cambridge, UK, 1993."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"R. Li, Z. Chen, and W. Wu,\n                      Generalized Difference Methods for Differential Equations\n                      , Marcel Dekker, New York, 2000. puncuate","DOI":"10.1201\/9781482270211"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"S. A. Orszag and M. Israeli,\n                      Numerical simulation of viscous incompressible flows\n                      , in Annual Review of Fluid Mechanics, Vol. 6, M. Van Dyke, W. A. Vincenti, and J. V. Wehausen, eds., Annual Reviews, Palo Alto, CA, 1974, pp. 281\u2013318.","DOI":"10.1146\/annurev.fl.06.010174.001433"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"M. J. D. Powell,\n                      Approximation Theory and Methods\n                      , Cambridge University Press, Cambridge, UK, 1981.","DOI":"10.1017\/CBO9781139171502"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(84)90015-9"},{"key":"R20","unstructured":"J. Strikwerda,\n                      Finite Difference Schemes and Partial Differential Equations\n                      , Wadsworth and Brooks\/Cole, Pacific Grove, CA, 1989."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/05062915X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:53:43Z","timestamp":1787327623000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/05062915X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":20,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/05062915X"],"URL":"https:\/\/doi.org\/10.1137\/05062915x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}