http://arxiv.org/abs/1804.10320
This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as \textit{grad} and \textit{div}. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.
G. Vasil, D. Lecoanet, K. Burns, et. al.
Mon, 30 Apr 18
-122/63
Comments: Submitted to JCP simultaneously with Part-II