## Abstract

This paper presents a method for accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The method 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 gradient and divergence. 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.

Original language | English (US) |
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Article number | 100013 |

Journal | Journal of Computational Physics: X |

Volume | 3 |

DOIs | |

State | Published - Jun 2019 |

## Keywords

- Coordinate singularities
- Jacobi polynomials
- Sparse operators
- Spectral methods
- Spherical geometry
- Spin-weighted spherical harmonics

## ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Computer Science Applications