Transport schemes in spherical geometries using spline-based RBF-FD with polynomials

Abstract

This work presents a numerical algorithm for using radial basis function-generated finite differences (RBF-FD) to solve partial differential equations (PDEs) on S2 using polyharmonic splines with added polynomials defined in a 2D plane (PHS+Poly). We introduce a novel method for calculating RBF-FD PHS+Poly differentiation weights on S2 using first a Householder reflection and then a projection onto the tangent plane. The new PHS+Poly RBF-FD method is implemented on two standard test cases: 1) solid body rotation on S2 and 2) 3D tracer transport within Earth's atmosphere. Compared to existing methods (including those in the RBF literature) at similar resolutions, this approach requires fewer degrees of freedom and is algorithmically much simpler. A MATLAB code to implement the method is included in the Appendix.