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An adaptive time-step allows one to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed, on the one hand, to analyze different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyze steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional (2d and 3d) physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations.<\/jats:p>","DOI":"10.1137\/080718619","type":"journal-article","created":{"date-parts":[[2010,5,26]],"date-time":"2010-05-26T18:52:18Z","timestamp":1274899938000},"page":"484-518","source":"Crossref","is-referenced-by-count":28,"title":["Time Integration and Steady-State Continuation for 2d Lubrication Equations"],"prefix":"10.1137","volume":"9","author":[{"given":"Philippe","family":"Beltrame","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Uwe","family":"Thiele","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1021\/ie049640r"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1021\/la900635f"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1038\/nmat788"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"P. 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Knobloch,\n                      On the depinning of a driven drop on a heterogeneous substrate\n                      , New J. Phys., 8 (2006), paper 313.","DOI":"10.1088\/1367-2630\/8\/12\/313"},{"key":"R93","doi-asserted-by":"crossref","unstructured":"U. Thiele, I. Vancea, A. J. Archer, M. J. Robbins, L. Frastia, A. Stannard, E. Pauliac-Vaujour, C. P. Martin, M. O. Blunt, and P. J. Moriarty,\n                      Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions\n                      , J. Phys. Condens. Matter, 21 (2009), paper 264016.","DOI":"10.1088\/0953-8984\/21\/26\/264016"},{"key":"R94","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112005007007"},{"key":"R95","doi-asserted-by":"crossref","unstructured":"U. Thiele, M. G. Velarde, and K. Neuffer,\n                      Dewetting: Film rupture by nucleation in the spinodal regime\n                      , Phys. Rev. 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Barkley,\n                      Bifurcation analysis for timesteppers\n                      , in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Vol. Math. Appl. 119, E. Doedel and L. 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