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An RBF-FD closest point method for solving PDEs on surfaces. (English) Zbl 1395.65029

Summary: Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method [the third author and B. Merriman, ibid. 227, No. 3, 1943–1961 (2008; Zbl 1134.65058)] 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 [C. Piret, ibid. 231, No. 14, 4662–4675 (2012; Zbl 1248.35009)], 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 [S. Leung and H. Zhao, ibid. 228, No. 8, 2993–3024 (2009; Zbl 1161.65013)]. 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R01 PDEs on manifolds

Software:

OEIS; rbf_qr; Matlab
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Full Text: DOI arXiv Link

References:

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