Natasha Flyer
Dept. of Applied Mathematics
University of Colorado
Boulder, Colorado, USA
Since the early 1990s, radial basis functions (RBFs) have been applied to solve simple elliptical and mixed parabolic partial differential equations (PDEs). Not until 2003 was the RBF methodology applied to spherical geometries for such PDEs, and then only to low algebraic accuracy. In 2007, the first study to successfully apply RBFs to numerically integrating hyperbolic PDEs on a sphere was done (with spectral accuracy easily being achieved). Now, the first RBF calculation to: 1) solve the nonlinear shallow water equations on the sphere for both steady and unsteady flow as well as 2) use mesh-free local node refinement with RBFs is presented. For the latter, the test case used is the wrap-up of two moving vortices being advected on the surface of the sphere at an arbitrary angle. This is a particularly challenging test due to the sharp gradients, with local areas of fine detail demanding high resolution. The RBF methodology is evaluated in terms of accuracy and time stability. Comparisons with common spectral methods as well as a finite volume method for the latter test will be given.