Periodized Radial Basis Functions (RBFs) and RBF-Vortex Method for the Barotropic Vorticity Equation.

Abstract

Fluids spontaneously develop fronts, narrow spiral filaments and other features of rapid spatial variation which are very challenging for numerical methods. Like most competing numerical schemes, Radial Basis Function (RBF) methods are based on interpolation. It has been previously proved that the RBF approximation converges to the correct solution as the number of grid points increases. When the flow is varying rapidly, high accuracy requires a high density of interpolation points while smooth regions require a lower density of points. A method that can adaptively allocate more grid points to where the fronts develop and fewer grid points to where the flow is smooth is of great value in fluid simulation on the surface of a sphere. In this thesis, a method that combines the meshfree nature of RBF interpolation and the Lagrangian particle method is developed. On the one hand, the particles serving as fluid elements are advected by the velocity field such that rapidly varying regions are densely populated; on the other hand, the particles serving as RBF centers provide higher density of interpolation points and therefore give a better resolution of the regions.