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Full-Text Articles in Physical Sciences and Mathematics
Node Generation For Rbf-Fd Methods By Qr Factorization, Tony Liu, Rodrigo B. Platte
Node Generation For Rbf-Fd Methods By Qr Factorization, Tony Liu, Rodrigo B. Platte
Faculty Publications
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 …
A Radial Basis Function Finite Difference Scheme For The Benjamin–Ono Equation, Benjamin F. Akers, Tony Liu, Jonah A. Reeger
A Radial Basis Function Finite Difference Scheme For The Benjamin–Ono Equation, Benjamin F. Akers, Tony Liu, Jonah A. Reeger
Faculty Publications
A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on ℝ, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.