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

4 Scopus citations


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
StatePublished - Jun 2019
Externally publishedYes


  • 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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