Seminars & Events

Mathematics and Computer Science Division
"A Cartesian Treecode for Multiquadric Radial Basis Functions and a Vortex Method for Fluid Flow on a Sphere"

DATE: July 28, 2010
TIME: 10:30 AM - 11:30 AM
SPEAKER: Lei Wang, University of Michigan, LANS Postdoc Interviewee
LOCATION: Building 240 Seminar Room 4301, Argonne National Laboratory
HOST: Mihai Anitescu

Description:
This talk concerns particle methods and has two parts. First we present a treecode algorithm for evaluating multiquadric radial basis function (RBF) approximations. The treecode employs a divide and conquer strategy and replaces particle-particle interactions by particle-cluster interactions which are efficiently computed by a far-field Cartesian Taylor approximation. For the multiquadric RBF, $\phi(x) = \sqrt{x^2 + c^2}$, the Laurent series presented in the literature converges only for a limited range of the RBF shape parameter $0 \le c \le c^*$, but the Taylor series employed here converges for all $c \ge 0$. The treecode reduces the computational cost from $O(N^2)$ to $O(N\log N)$ operations, where $N$ is the size of the system. Second we discuss work in progress on solving the Barotropic Vorticity Equation on a rotating sphere by a vortex method. Vortex methods are Lagrangian techniques in contrast with more conventional Eulerian methods such as finite difference, finite element and spectral methods. The vortex method uses Lagrangian particles and panels to track the flow map and absolute vorticity. The velocity is computed from the Biot-Savart integral on the sphere. An adaptive refinement strategy is implemented to resolve small-scale features. Results are presented for Rossby-Haurwitz waves and vortex patch interactions. The two parts of the talk are related because the treecode can be used to evaluate the Biot-Savart integral on the sphere.

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