Tensor calculus in spherical coordinates using Jacobi polynomials, Part-II: Implementation and Examples

TL;DR

This paper introduces a spectral simulation code for solving tensorial partial differential equations in a sphere, demonstrating its accuracy and flexibility through various benchmark problems and detailed implementation.

Contribution

The paper presents a novel spectral method implementation for tensor PDEs in spherical coordinates, including code, benchmarks, and analysis of numerical sensitivities.

Findings

The code accurately solves linear tensor PDEs and implements boundary conditions.

Higher resolution and order schemes improve solution accuracy.

Small numerical changes can significantly affect low-resolution dynamo simulations.

Abstract

We present a simulation code which can solve broad ranges of partial differential equations in a full sphere. The code expands tensorial variables in a spectral series of spin-weighted spherical harmonics in the angular directions and a scaled Jacobi polynomial basis in the radial direction, as described in Part-I. Nonlinear terms are calculated by transforming from the coefficients in the spectral series to the value of each quantity on the physical grid, where it is easy to calculate products and perform other local operations. The expansion makes it straightforward to solve equations in tensor form (i.e., without decomposition into scalars). We propose and study several unit tests which demonstrate the code can accurately solve linear problems, implement boundary conditions, and transform between spectral and physical space. We then run a series of benchmark problems proposed in Marti et al (2014), implementing the hydrodynamic and magnetohydrodynamic equations. We are able to calculate more accurate solutions than reported in Marti et al 2014 by running at higher spatial resolution and using a higher-order timestepping scheme. We find the rotating convection and convective dynamo benchmark problems depend sensitively on details of timestepping and data analysis. We also demonstrate that in low resolution simulations of the dynamo problem, small changes in a numerical scheme can lead to large changes in the solution. To aid future comparison to these benchmarks, we include the source code used to generate the data, as well as the data and analysis scripts used to generate the figures.