Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

Geoffrey M. Vasil*, Daniel Lecoanet, Keaton J. Burns, Jeffrey S. Oishi, Benjamin P. Brown

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

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 languageEnglish (US)
Article number100013
JournalJournal of Computational Physics: X
Volume3
DOIs
StatePublished - Jun 2019

Funding

GMV acknowledges support from the Australian Research Council , project number DE140101960 . DL is supported by a Hertz Foundation Fellowship, the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400 , a PCTS fellowship, and a Lyman Spitzer Jr. fellowship. BPB and JSO were partially supported in this work by NASA LWS grant NNX16AC92G . We thank Keith Julien, Sheehan Olver, and Alex Townsend for many useful conversations and insights regarding sparse spectral methods. We also thank an anonymous referee for an extremely careful reading of the manuscript and numerous helpful improvement suggestions.

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

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