In this paper, we present a method based on Radial Basis Function(RBF)-generated Finite Differences (FD) for numerically solving diffusion andreaction-diffusion equations (PDEs) on closed surfaces embedded in$\mathbb{R}^d$. Our method uses a method-of-lines formulation, in which surfacederivatives that appear in the PDEs are approximated locally using RBFinterpolation. The method requires only scattered nodes representing thesurface and normal vectors at those scattered nodes. All computations use onlyextrinsic coordinates, thereby avoiding coordinate distortions andsingularities. We also present an optimization procedure that allows for thestabilization of the discrete differential operators generated by our RBF-FDmethod by selecting shape parameters for each stencil that correspond to aglobal target condition number. We show the convergence of our method on twosurfaces for different stencil sizes, and present applications to nonlinearPDEs simulated both on implicit/parametric surfaces and more general surfacesrepresented by point clouds.
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机译：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，用于数值求解嵌入在\\ mathbb {R} ^ d $中的闭合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插值法对PDE中出现的表面衍生物进行局部近似。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向矢量。所有计算仅使用外部坐标，从而避免坐标变形和奇异性。我们还提出了一种优化程序，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。我们展示了针对不同模具尺寸的两个表面上我们的方法的收敛性，并介绍了在隐式/参数化表面以及由点云表示的更通用表面上模拟的非线性PDE的应用。
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