Fourth order partial differential equations on general geometries

We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces [M. Bertalmío, L.T. Cheng, S. Osher, G. Sapiro. Variational problems and partial differential 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 R^N as the level set of a smooth function, φ{symbol}, we compute the PDE using only finite differences on a standard Cartesian mesh in R^N. The higher order equations introduce a number of challenges that are of less concern when applying this method to first 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 difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn-Hilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.