Rotational transport on a sphere: Local node refinement with radial basis functions q Natasha Flyer a, * , Erik Lehto b a Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO 80305, USA b Division of Scientific Computing, Uppsala University, Sweden article info Article history: Received 18 March 2009 Received in revised form 9 November 2009 Accepted 10 November 2009 Available online 16 November 2009 Keywords: Radial basis functions Local node refinement Hyberbolic PDEs abstract This paper develops an algorithm for radial basis function (RBF) local node refinement and implements it for vortex roll-up and transport on a sphere. A heuristic based on an electro- static repulsion type principle is used to re-distribute the nodes, clustering in areas where higher resolution is needed. It is then important to have a scheme that varies the shape of the RBFs over the domain so as to counteract the effects of Runge phenomena where the nodes are sparse. The roll-up of two diametrically opposed moving vortices are studied. The performance differences between near-uniform and refined nodes are addressed in terms of convergence, time stability, and computational cost. RBF results are put into context by comparison with published results for methods such as finite volume and discontinuous Galerkin. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Although applications of radial basis functions (RBFs) have bloomed in recent years, using RBFs to solve evolutionary par- tial differential equations (PDEs) is a young research field. The strength of the method is in its ability to achieve spectral or high-order accuracy for scattered node layouts while being able to node refine in areas where increased resolution is needed. Although, this latter quality of local node refinement seems to naturally extend from the method being meshless and thus being able to place the points where needed, few papers have addressed this issue and the numerical complications that arise in doing so [6,14,16]. It is the aim of this paper to develop a meshless algorithm for RBF local node refinement on the sphere. Since physical phenomena in fluid dynamics often require variable resolution depending on the formation of flow fea- tures, it would be advantageous to have a method that would naturally refine according to the physics. Currently used meth- ods that allow for local mesh refinement, such as finite volume or elements, discontinuous Galerkin, and spectral elements, are linked to underlying grids that introduce artificial boundaries necessary to perform the numerics. In contrast, since RBFs are not linked to any surface-based coordinate system (i.e. grid or mesh), the placement of the nodes and how they are re- fined will physically reflect the features of the flow (and not resemble boxes, triangles, etc.). However, one can not simply ‘clump’ where needed without taking into account the Runge phenomenon, ill-conditioning, and adverse effects on the smoothness of the solution. The question then arises, ‘How does one node refine?’. Since even without boundaries these issues arise, we will begin with defining a methodology for local node refinement in periodic domains, such as the surface of a sphere. 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.11.016 q The National Center for Atmospheric Research is sponsored by the National Science Foundation. The work was supported by NSF Grant ATM-0620100. * Corresponding author. Address: Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO 80305, USA. Tel.: +1 303 497 1292. E-mail addresses: flyer@ucar.edu (N. Flyer), erik.lehto@it.uu.se (E. Lehto). Journal of Computational Physics 229 (2010) 1954–1969 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp