This paper presents a conservative finite difference scheme for solving the N-component Cahn–Hilliard (CH) system on curved surfaces in three-dimensional (3D) space. Inspired by the closest point method (Macdonald and Ruuth, SIAM J Sci Comput 31(6):4330–4350, 2019), we use the standard seven-point finite difference discretization for the Laplacian operator instead of the Laplacian–Beltrami operator. We only need to independently solve ($$N-1$$) CH equations in a narrow band domain around the surface because the solution for the Nth component can be obtained directly. The N-component CH system is discretized using an unconditionally stable nonlinear splitting numerical scheme, and it is solved by using a Jacobi-type iteration. Several numerical tests are performed to demonstrate the capability of the proposed numerical scheme. The proposed multicomponent model can be simply modified to simulate phase separation in a complex domain on 3D surfaces.

中文翻译：

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3D 曲面上 N 分量 Cahn-Hilliard 系统的保守有限差分格式

本文提出了一种保守的有限差分方案，用于求解三维 (3D) 空间曲面上的 N 分量 Cahn-Hilliard (CH) 系统。受最近点法（Macdonald and Ruuth, SIAM J Sci Comput 31(6):4330–4350, 2019）的启发，我们对拉普拉斯算子使用标准的七点有限差分离散化，而不是拉普拉斯-贝尔特拉米算子。我们只需要在表面周围的窄带域中独立求解 ($$N-1$$) CH 方程，因为可以直接获得第 N 个分量的解。N 分量 CH 系统使用无条件稳定非线性分裂数值方案进行离散化，并使用 Jacobi 型迭代求解。进行了几次数值试验以证明所提出的数值方案的能力。