An RBF-FD closest point method for solving PDEs on surfaces

Highlights: • In this paper, a new method for the numerical approximation of PDEs on surfaces is proposed. Our method has the advantage of being comprised of standard computational components, such as the closest point representation of the surface, and RBF finite difference methods. • Our approach uses a narrow computational tube around the surface and avoids the need for a quasi-uniform distribution of surface points. This makes the method a natural candidate for coupling with grid-based methods such as the grid-based particle method for moving interface problems (Leung and Zhao, J. Comput. Phys. 228 (8) (2009) 2993–3024). • The method is also efficient: it exploits repeated patterns in computational geometry, it uses small computational tubes, and it avoids an explicit interpolation step. Further-more, a change in the order of the method is carried out simply by changing the number of points in the finite difference stencil. See our novelty statement for details on how the method compares with the original closest point method (Ruuth and Merriman, J. Comput. Phys. 227 (3) (2008) 1943–1961) and recent RBF methods (e.g., Piret, J. Comput. Phys. 231 (14) (2012) 4662–4675). • We conduct convergence studies in two and three dimensions and apply the method to a variety of problems, including reaction–diffusion systems and image denoising. Second order accurate results are observed in our experiments. Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008) [17]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012) [22]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009) [26]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.