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

[摘要] In this thesis we investigate three related topics involving the accuracy and efficiency ofnumerical 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 interpolatingmultidimensional scattered data and it also hasgreat 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 fastCartesian treecode for evaluating RBFs efficiently. The method applies a divide andconquer 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-scalehorizontal motions of the atmosphere. We first review the basic properties and analytic solutions of the BVEand then give two approachesto solving the BVE numerically. The first one uses Gaussian RBFs and the second one uses the vortex method. Both methodssolve 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.

[发布机构] University of Michigan

[效力级别] Barotropic Vorticity Equation

[关键词] Lagrangian Vortex Method;Barotropic Vorticity Equation;Fast Treecode Algorithm;Radial Basis Function Method;Mathematics;Science;Applied and Interdisciplinary Mathematics