A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

Abstract

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.