In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝ d . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

中文翻译：

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用于表面扩散和反应扩散方程的径向基函数（RBF）-有限差分（FD）方法。

在本文中，我们提出了一种基于径向基函数（RBF）生成的有限差分（FD）的方法，用于数值求解embedded d中封闭表面上的扩散和反应扩散方程（PDE）。我们的方法采用线法公式化，其中使用RBF插值对PDE中出现的表面导数进行局部近似。该方法仅需要代表表面的分散节点和那些分散节点处的法线矢量。所有计算仅使用外部坐标，从而避免了坐标变形和奇异性。我们还提出了一种优化程序，该程序可以通过为每个模板选择与全局目标条件编号相对应的形状参数，来稳定由我们的RBF-FD方法生成的离散差分算子。