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

Abstract

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 spinweighted 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 grad and 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.

Dedalus: A flexible framework for numerical simulations with spectral methods

“…We use 256 × 128 Fourier modes in the x, y directions, and 32 Chebyshev modes in the z direction, all using 3/2 dealiasing. The Chebyshev tau method balances energy to exponentially high accuracy; exact energy conservation could be still be imposed, however, by formulating a self-adjoint system with an alternative orthogonal polynomial basis [50,98,99].…”

“…The flow vanishes at the outer wall; u(t, R out , θ) = 0. The tangential velocity at the inner wall is a time-dependent regularized square wave u θ (t, R in , θ) = arctan(50 sin(t/n rot )) arctan (50) .…”

Linearly forced fluid flow on a rotating sphere

“…The dynamics at the poles are similar to those on a non-rotating flat plane and the non-inertial effects matter the most at the equator. method with a basis of spin-weighted spherical harmonics Vasil et al 2019); more details on the method can be found in the appendix of Mickelin et al (2018). A spectral expansion with a cut-off max = 256 was found to be sufficient to obtain converged solutions.…”

A systematic approach to reduce the independent tensor components by symmetry transformations: a commented translation of “Tensors and Crystal Symmetry” by Carl Hermann

“…This complex basis has the nice property that rotations by π/2 around the vertical axis are obtained by multiplying A 1 and A 2 by the imaginary unit i, which is conveniently summarized with the components while the basis is unaffected; see e.g. Vasil et al [22].…”

“…For the threefold, fourfold and sixfold axes, the following component vanishes C 11 12 = 1 4 (c 11 11 − c 22 22 − c 12 12 + c 11 22 + ic 11 21 + ic 22 12 + c 21 22 − c 22 11 − ic 11 12 − ic 22…”