RBF-vortex methods for the barotropic vorticity equation on a sphere

Vortex blob methods approximate a flow as a sum of many small vortices of Gaussian shape, and adaptively move the vortex centers with the current. Gaussian radial basis functions (RBFs) do exactly the same. However, RBFs solve an exact interpolation problem - expensive but accurate - while vortex methods sacrifice accuracy through quasi-interpolation for the absence of a matrix inversion. We show that vortex-RBF algorithms with spectral accuracy are stable for flows on the sphere. The version in Eulerian coordinates is fast; the fully-Lagrangian variant is much slower for a given basis size N, but is highly adaptive for advection-dominated flows. Both versions are excellent for small-to-medium-N problems - N up to 10,000, say, where N is the number of RBF grid points/vortex blobs. Neither is good for large N problems because the cost of the Eulerian model scales as N2 per timestep while the Lagrangian vortex-RBF method scales as N3. The slow-Lagrangian scheme is unique among vortex methods in being genuinely (like its Eulerian sibling) a spectrally-accurate method: for laminar flows the error falls exponentially fast with N.