Coast to Coast Seminar Series: "Fourier Spectral Computing on the Sphere"

Abstract

Although spherical coordinates arise naturally in many applications, numerical routines for computing PDEs on a spherical surface are not yet commonplace tools for the relatively uninitiated. While spherical harmonics and finite-element methods are well-developed approaches, neither possesses the essential simplicity of computing on the 2D periodic domain with FFT-based spectral methods. It is little appreciated that fast Fourier transforms for the spherical surface have been implemented using the fact that longitude-latitude coordinates can be double-mapped to the torus. Combining this idea with a choice of Fourier basis for which the Laplacian is a sparse matrix operation, implicit time-stepping for diffusion is implemented in a spectrally-fast manner.

In this Coast-to-Coast presentation, the elementary ideas behind this simple FFT-based approach to PDE computations on the sphere will be illustrated with a series of MATLAB demo codes. The codes will be made available prior to the seminar, and participants are highly encouraged to run their own tests during the session. The fast algorithms will allow reasonably resolved computations to execute in minutes on even a basic laptop computer (with the MATLAB software installed). This spectral method is applied to several examples of diffusion-driven dynamics in models of pattern formation.