Efficient spectral methods for partial differential equations in spherical domain

Lina Ma, Purdue University

Abstract

There are many important problems related to spherical domains. Most common examples would be the Earth, and other astrophysical bodies as the Sun. What we want is to use spherical coordinate to represent the spherical domain and expand scalar/vector functions with the help of spherical harmonics functions. By performing spherical harmonic analysis, we can turn global operators into local operators, and reduce dimensionality of the problem, therefore increase computational efficiency and accuracy by applying the high-order spectral methods later. The problems we are going to study in this thesis are exterior problem of Maxwell Equations, Magneto-Hydrodynamic on the Sun, Primitive Equations of the Atmosphere, and Boussineq Equations. These problems have many practical uses in radar detecting, bio-imaging, space weather predicting, and atmospheric and oceanic circulations study. The main purpose of this thesis is to show how to perform spherical harmonic analysis in these problems, especially turn the complicated high dimension vector problems into a sequence of one dimensional linear system. We then design accordingly the numerical scheme with the help of spectral-Galerkin methods and show the exponential convergency in numerical solution.

Degree

Ph.D.

Advisors

Shen, Purdue University.

Subject Area

Mathematics

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