An ALE meshfree method for surface PDEs coupling with forced mean curvature flow

Abstract

In this paper, we propose an arbitrary Lagrangian Eulerian (ALE) meshfree method that utilizes the radial basis function-finite difference (RBF-FD) method for solving diffusion-reaction equations on evolving surfaces. The surface evolution law is determined by a forced mean curvature flow (FMCF). We develop a parametric RBF-FD method to solving the FMCF, which has not been done previously in the framework of meshfree methods. The advantage of this parametric method is that it utilizes a suitable tangential velocity in the equation to maintain a fairly uniform distribution of nodes during the surface evolution. The tangential velocity also allows us to construct an ALE RBF-FD scheme for the surface PDEs, which performs better than the purely Lagrangian method. Furthermore, we discuss how to deal with the boundary conditions for both the FMCF and the PDEs on surfaces, and present a ghost node approach to efficiently discretize the boundary conditions. Some numerical experiments are shown, which not only demonstrate the effectiveness of the tangential velocity, but also illustrate two applications.