A Robust RBF-FD Formulation based on Polyharmonic Splines and Polynomials

Gregory Allen Barnett, University of Colorado at Boulder


We introduce a local method based on radial basis function-generated finite differences (RBFFD) for interpolation and the numerical solution of partial differential equations (PDEs). The method uses polyharmonic spline (PHS) RBFs together with polynomials to derive differentiation weights on different node configurations. The formulation is explored in three directions: (i) Interpolation and approximation of differential operators, (ii) Elliptic PDEs, and (iii) Hyperbolic PDEs. In particular, the novel RBF-FD methodology is applied to standard test cases in numerical weather prediction, modeled by the compressible Navier-Stokes equations in 2D. Furthermore, the evaluation of the method on different node layouts, Cartesian, hexagonal, and scattered, is studied. The RBF-FD implementation acts as an extension of conventional finite-differences, achieving high accuracy on scattered nodes with no need for a computational mesh.