Stabilized Radial Basis Function Finite Difference Schemes with Mass Conservation for the Cahn–Hilliard Equation on Surfaces

It is well known that the Cahn-Hilliard equation satisfies the energy dissipation law and the mass conservation property. Recently, the radial basis function-finite difference (RBF-FD) approach and its numerous variants have garnered significant attention for the numerical solution of surface-related problems, owing to their intrinsic advantage in handling complex geometries. However, existing RBF-FD schemes generally fail to preserve mass conservation when solving the Cahn-Hilliard equation on smooth closed surfaces. In this paper, based on an L2 projection method, two numerically efficient RBF-FD schemes are proposed to achieve mass-conservative numerical solutions, which are demonstrated to preserve the mass conservation law under relatively mild time-step constraints. Spatial discretization is performed using the RBF-FD method, while based on the convex splitting method and a linear stabilization technique, the first-order backward Euler formula (BDF1) and the second-order Crank-Nicolson (CN) scheme are employed for temporal integration. Extensive numerical experiments not only validate the performance of the proposed numerical schemes but also demonstrate their ability to utilize mild time steps for long-term phase-separation simulations.