Tensor calculus in spherical coordinates using Jacobi polynomials. I: Mathematical analysis and derivations. (English) Zbl 07785508
Summary: 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.
MSC:
| 65Lxx | Numerical methods for ordinary differential equations |
| 65Dxx | Numerical approximation and computational geometry (primarily algorithms) |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
spherical geometry; coordinate singularities; spectral methods; spin-weighted spherical harmonics; Jacobi polynomials; sparse operatorsReferences:
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