First- and second-order unconditionally stable direct discretization methods for multi-component Cahn–Hilliard system on surfaces

This paper proposes a first- and second-order unconditionally stable direct discretization method based on a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation for solving the N-component Cahn–Hilliard system. We define the discretizations of the gradient, divergence, and Laplace–Beltrami operators on triangle surfaces. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. We also prove that our method satisfies the mass conservation. The proposed scheme is solved by the biconjugate gradient stabilized (BiCGSTAB) method, which can be straightforwardly applied to GPU-accelerated biconjugate gradient stabilized implementation by using the Matlab Parallel Computing Toolbox. Several numerical experiments are performed and confirm the accuracy, stability, and efficiency of our proposed algorithm.