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Fourth Order Partial Differential Equations



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Journal of Computational Physics 216 2006 216 246 www elsevier com locate jcp Fourth order partial di erential equations on general geometries John B Greer a a Andrea L Bertozzi b Guillermo Sapiro c Department of Mathematics Courant Institute for Mathematical Sciences New York University 251 Mercer Street New York NY 10012 1185 United States b Department of Mathematics UCLA Los Angeles CA 90095 United States c Electrical and Computer Engineering University of Minnesota Minneapolis MN 55455 United States Received 22 March 2005 received in revised form 3 November 2005 accepted 30 November 2005 Available online 23 January 2006 Abstract We extend a recently introduced method for numerically solving partial di erential equations on implicit surfaces M Bertalm o L T Cheng S Osher G Sapiro Variational problems and partial di erential equations on implicit surfaces J Comput Phys 174 2 2001 759 780 to fourth order PDEs including the Cahn Hilliard equation and a lubrication model for curved surfaces By representing a surface in RN as the level set of a smooth function we compute the PDE using only nite di erences on a standard Cartesian mesh in RN The higher order equations introduce a number of challenges that are of less concern when applying this method to rst and second order PDEs Many of these problems such as time stepping restrictions and large stencil sizes are shared by standard fourth order equations in Euclidean domains but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry We approach these di culties by applying convexity splitting methods ADI schemes and iterative solvers We discuss in detail the di erences between computing these fourth order equations and computing the rst and second order PDEs considered in earlier work We explicitly derive schemes for the linear fourth order di usion the Cahn Hilliard equation for phase transition in a binary alloy and surface tension driven ows on complex geometries



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