Abstract
Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.
Highlights
In [1,2,3,4,5,6], Polyharmonic Splines (PHSs) and polynomials were combined to generate radial basis function finite-difference (RBF-FD) methods
One of the key benefits of combining PHSs with polynomials was the fact that high-order accuracy could be obtained from resulting radial basis functions (RBFs)-FD differentiation matrices
To find the RBF-FD weights for a given operator L, we first consider the system with strictly RBFs, as shown in Equation (4)
Summary
In [1,2,3,4,5,6], Polyharmonic Splines (PHSs) and polynomials were combined to generate radial basis function finite-difference (RBF-FD) methods. One of the key benefits of combining PHSs with polynomials was the fact that high-order accuracy could be obtained from resulting RBF-FD differentiation matrices Another improvement was the elimination of the requirement to select optimal shape parameters. These methods maintain accurate approximations while eliminating the complexities of shape parameter selection Along with these advantages for using PHSs with polynomials came one key constraint: the number of nodes used in each stencil was required to be approximately twice the size of the number of polynomial basis functions appended. The sampled points mitigate a key computational constraint of RBF-FD methods implemented with PHSs and polynomials That is, it dramatically reduces the number of nodes per stencil for high-order approximation as compared to other node distributions such as Cartesian or hexagonal points.
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