RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for Solving PDEs on Surfaces
RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for Solving PDEs on Surfaces
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DOI：
10.1016/j.jcp.2018.07.015
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发表时间：
2018-07
期刊：
影响因子：
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通讯作者：
Varun Shankar;A. Narayan;R. Kirby
中科院分区：
文献类型：
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作者：
Varun Shankar;A. Narayan;R. Kirby
提出了一种利用稳定的无标度径向基函数（RBF）插值求解余维流形M ∈ Rd上偏微分方程的新方法.我们的方法涉及增加多调和样条（PHS）径向基函数与多项式生成径向基函数有限差分（RBF-FD）公式。这些多项式基元使用最近开发的最小正交插值技术（LOI）在每个RBF-FD模板上获得，以获得R3中的多项式对M上的模板的局部限制。由此产生的RBF-LOI方法使用笛卡尔坐标，不需要任何内在的坐标系或投影点到切平面上，我们的测试表明了稳定性的错误。我们表明，我们的方法产生高阶的收敛性的球和环面上的偏微分方程，并提出了一些应用生物学的反应扩散偏微分方程。
We present a new method for the solution of PDEs on manifolds M⊂ R d of co-dimension one using stable scale-free radial basis function (RBF) interpolation. Our method involves augmenting polyharmonic spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD) formulas. These polynomial basis elements are obtained using the recently-developed least orthogonal interpolation technique (LOI) on each RBF-FD stencil to obtain local restrictions of polynomials in R 3 to stencils on M. The resulting RBF-LOI method uses Cartesian coordinates, does not require any intrinsic coordinate systems or projections of points onto tangent planes, and our tests illustrate robustness to stagnation errors. We show that our method produces high orders of convergence for PDEs on the sphere and torus, and present some applications to reaction–diffusion PDEs motivated by biology.