Numerical computation of advection and diffusion on evolving diffuse interfaces

We propose a numerical method for computing transport and diffusion on a moving surface. The approach is based on a diffuse interface model in which a bulk diffusion–advection equation is solved on a layer of thickness ϵ containing the surface. The conserved quantity in the bulk domain is the concentration weighted by a density which vanishes on the boundary of the thin domain. Such a density arises naturally in double obstacle phase field models. The discrete equations are then formulated on a moving narrow band consisting of grid points on a fixed mesh. We show that the discrete equations are solvable subject to a natural constraint on the evolution of the discrete narrow band. Mass is conserved and the discrete solution satisfies stability bounds. Numerical experiments indicate that the method is second-order accurate in space.