| Abstract: |
| 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 a closest point representation of the surface, a constant-along-normal extension is employed to formulate the PDE in the embedded space, which can be solved numerically using standard finite difference schemes.
We present a closest point method that uses finite difference schemes derived from radial basis functions (RBF-FD). When compared to the standard finite difference discretization of the original 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. Our method uses RBF centers on regular grid nodes, avoiding the ill-conditioning from point clustering on the surface. An implicit formulation that uses the least-squares method allows an easy and natural coupling with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). The method is tested in a number of applications on static and moving surfaces. |
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