The closest point method for the numerical solution of partial differential equations on moving surfaces

Partial differential equations (PDEs) on surfaces arise in a wide range of applications. The closest point method is a recent embedding method that has been used to solve a variety of PDEs on smooth surfaces using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. The original closest point method (CPM) was designed for problems posed on static surfaces, however the solution of PDEs on moving surfaces is of considerable interest as well. Here we propose two different approaches for solving PDEs on moving surfaces using a combination of the CPM and a grid based particle method. The grid based particle method (GBPM) represents and tracks surfaces using meshless particles and an Eulerian reference grid. In our first approach, a modification of the GBPM is introduced to ensure that all the grid points within a computational tube surrounding the surface are active. The modified GBPM is tested in geometric motions of surfaces to verify the correctness of the new algorithm. A coupled method is proposed combining the modified GBPM and the CPM and tested on a number of numerical examples. Our second approach uses generalized finite difference schemes derived from radial basis functions (RBF-FD) in the implementation of the closest point method. An explicit and an implicit formulation of the CPM using RBF-FD are presented along with numerical experiments for the convergence of the method, including the approximation of the solution of reaction-diffusion equations and the Cahn-Hilliard equation on a variety of surfaces. Finally, a coupled method of the CPM that uses RBF-FD and the original GBPM is proposed and used to solve PDEs on moving surfaces.