RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

JOURNAL OF COMPUTATIONAL PHYSICS

卷:373

Article

Shankar, Varun¹  Narayan, Akil^1,2  Kirby, Robert M.² 

[1] Univ Utah, Dept Math, Salt Lake City, UT 84112 USA

[2] Univ Utah, Sci Comp & Imaging Inst, Salt Lake City, UT 84112 USA

关键词: Radial basis function;    High-order method;    Manifolds;   

DOI  :  10.1016/j.jcp.2018.07.015

来源: Elsevier

【 摘 要 】

We present a new method for the solution of PDEs on manifolds M subset of 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free