Motion by surface diffusion is a type of surface-area-diminishing flow such that the enclosed volume is preserved and is important is many physical applications, including solid state de-wetting. In this talk I will describe a relatively recent diffuse interface model for surface diffusion, wherein the sharp-interface surface description is replaced by a diffuse interface, or boundary layer, with respect to some order parameter. One of the nice features of the new doubly degenerate Cahn-Hilliard (DDCH) model is that it permits a hyperbolic tangent description of the diffuse interfaces, in an asymptotic sense, but, at the same time, supports a maximum principle, meaning that the order parameter stays between two predetermined values. Furthermore, numerics show that convergence to the sharp interface solutions for the DDCH model is faster than that of the standard regular Cahn-Hilliard (rCH) model. The down side is that the new DDCH model is singular and much more nonlinear than the rCH model, which makes numerical solution difficult, and it is still only first order accurate asymptotically. We will describe positivity-preserving numerical methods for the new model and review some existing numerics. We will also describe very recent results on the rigorous Gamma convergence of the underlying diffuse interface energy.
Thackeray Hall 427