Tensor calculus in polar coordinates using Jacobi polynomials

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Original languageEnglish (US)
Pages (from-to)53-73
Number of pages21
JournalJournal of Computational Physics
StatePublished - Nov 15 2016
Externally publishedYes


  • Fluid mechanics
  • Jacobi polynomials
  • Numerical analysis
  • Orthogonal polynomials
  • Partial differential equations
  • Pipe flow

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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