Direct Discretization Method for the Cahn-Hilliard Equation on an Evolving Surface

We propose a simple and efficient direct discretization scheme for solving the Cahn-Hilliard (CH) equation on an evolving surface. By using a conservation law and transport formulae, we derive the CH equation on evolving surfaces. An evolving surface is discretized using an unstructured triangular mesh. The discrete CH equation is defined on the surface mesh and its dual surface polygonal tessellation. The evolving triangular surfaces are then realized by moving the surface nodes according to a given velocity field. The proposed scheme is based on the Crank-Nicolson scheme and a linearly stabilized splitting scheme. The scheme is second-order accurate, with respect to both space and time. The resulting system of discrete equations is easy to implement, and is solved by using an efficient biconjugate gradient stabilized method. Several numerical experiments are presented to demonstrate the performance and effectiveness of the proposed numerical scheme.