  Partial differential equations (PDEs) on surfaces appear in many applications
throughout the natural and applied sciences. The classical closest point method
(Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) 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, J. Comput. Phys. 231(14):4662-4675,
[2012]), 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, J.
Comput. Phys. 228(8):2993-3024, [2009]). 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.
