Radial Basis Functions and Vortex Methods and their Application to Vortex Dynamics on a Rotating Sphere.

Abstract

In this thesis we investigate three related topics involving the accuracy and efficiency of numerical algorithms for sicentific computation.

The first topic is on the Radial Basis Function (RBF) method. The RBF method is one of the primary tools for interpolating multidimensional scattered data and it also has great potential for solving Partial Differential Equations (PDEs). We develop an approximate cardinal function for the Gaussian RBF on an unbounded uniform grid in one dimension and compare to the Finite Difference (FD) method using Fourier analysis. We find that the truncated Gaussian RBF method is inferior to the FD for differentiating the function $f(X) = exp(iKX)$, where $K$ is the wavenumber.

The second topic is a fast Cartesian treecode for evaluating RBFs efficiently. The method applies a divide and conquer strategy and uses particle-cluster interactions in place of particle-particle interactions. Taylor approximation is applied for the far-field expansion. For multiquadric RBFs, $phi(x) = sqrt{x^2 + c^2}$, the Laurent series presented in the literature converges only for a limited range of $c$, but the Taylor series converges for all $cge0$. The treecode algorithm reduces the computational cost from $O(N^2)$ to $O(Nlog N)$ operations, where $N$ is the size of the system.

The third topic is the Barotropic Vorticity Equation (BVE), a simple model for the large-scale horizontal motions of the atmosphere. We first review the basic properties and analytic solutions of the BVE and then give two approaches to solving the BVE numerically. The first one uses Gaussian RBFs and the second one uses the vortex method. Both methods solve the BVE in a Lagrangian sense, that is, the particles are moving with the flow. In the vortex method, adaptive mesh refinement is used to track the small scale features. Rossby-Haurwitz waves and the evolution of Gaussian patches are investigated as numerical tests of both methods.