A level set approach for diffusion and Stefan-type problems with Robin boundary conditions on quadtree/octree adaptive Cartesian grids

Keywords: Level set method Epitaxial growth Diffusion Stefan problem Sharp interface Robin boundary condition a b s t r a c t We present a numerical method for simulating diffusion dominated phenomena on irregular domains and free moving boundaries with Robin boundary conditions on quadtree/ octree adaptive meshes. In particular, we use a hybrid finite-difference and finite-volume framework that combines the level-set finite difference discretization of Min and Gibou (2007) [13] with the treatment of Robin boundary conditions of Papac et al. (2010) [19] on uniform grids. We present numerical results in two and three spatial dimensions on the diffusion equation and on a Stefan-type problem. In addition, we present an application of this method to the case of the simulation of the Ehrlich–Schwoebel barrier in the context of epitaxial growth. Diffusion and Stefan-type problems with Robin boundary conditions are of practical significance in a variety of fields. For example, these equations arise in heat transfer applications involving internal conduction and convection [11]; tissue imaging with near-infrared tomography [21]; and continuum models of epitaxial growth involving the Ehrlich–Schwoebel barrier [28] on step edges.