A polynomial-augmented RBF collocation method using fictitious centres for solving the Cahn–Hilliard equation

Abstract

In this paper, we consider the radial basis function collocation method with fictitious centres for solving the Cahn–Hilliard equation in one-dimensional and two-dimensional settings. Radial basis functions with polynomial augmentation are used in derivative approximations to overcome the issue of the stagnation error and to avoid the hassle of determining the shape parameter. Meanwhile, we implement a semi-implicit technique for temporal discretization, which combines the stability of the implicit method and the simplicity of the explicit method. Numerical experiments show that the hybrid bases significantly improve the accuracy of the radial basis function collocation method and its stability in terms of the shape parameter c.