Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Physics and Astronomy

First Advisor

Joel E. Tohline


This dissertation addresses the need for an accurate and efficient technique which solves the Poisson equation for arbitrarily complex, isolated, self-gravitating fluid systems. Generally speaking, a potential solver is composed of two distinct pieces: a boundary solver and an interior solver. The boundary solver computes the potential, phi(xB) on a surface which bounds some finite volume of space, V, and contains an isolated mass-density distribution, rho(x). Given rho(x) and phi(xB), the interior solver computes the potential phi(x) everywhere within V. Herein, we describe the development of a numerical technique which efficiently solves Poisson's equation in cylindrical coordinates on massively parallel computing architectures. First, we report the discovery of a compact cylindrical Green's function (CCGF) expansion and show how the CCGF can be used to efficiently compute the exact numerical representation of phi(xB). As an analytical representation, the CCGF should prove to be extremely useful wherever one requires the isolated azimuthal modes of a self-gravitating system. We then discuss some mathematical consequences of the CCGF expansion, such as it's applicability to all nine axisymmetric coordinate systems which are R -separable for Laplace's equation. The CCGF expansion, as applied to the spherical coordinate system, leads to a second addition theorem for spherical harmonics. Finally, we present a massively parallel implementation of an interior solver which is based on a data-transpose technique applied to a Fourier-ADI (Alternating Direction Implicit) scheme. The data-transpose technique is a parallelization strategy in which all communication is restricted to global 3D data-transposition operations and all computations are subsequently performed with perfect load balance and zero communication. The potential solver, as implemented here in conjunction with the CCGF expansion, should prove to be an extremely useful tool in a wide variety of astrophysical studies, particularly those requiring an accurate determination of the gravitational field due to extremely flattened or highly elongated mass distributions.