A Radial Basis Function (rbf) Compact Finite

We present a new high-order, local meshfree method for numerically solving reaction 5 diffusion equations on smooth surfaces of co-dimension one embedded in Rd. The novelty of the 6 method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite 7 radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact 8 (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, 9 which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes 10 on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the 11 method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. 12 We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas 13 that can guarantee the resulting differentiation matrices have desirable stability properties. The 14 improved accuracy and computational cost that can be achieved with this method over the standard 15 (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate 16 the flexibility and general applicability of the method by solving two different reaction diffusion 17 equations on surfaces that are defined implicitly and only by point clouds. 18