We present a fully spectral methodology for magnetohydrodynamic (MHD) calculations in a whole sphere. The use of Jones-Worland polynomials for the radial expansion guarantees that the physical variables remain infinitely differentiable throughout the spherical volume. Furthermore, we present a mathematically motivated and systematic strategy to relax the very stringent time step constraint that is present close to the origin when a spherical harmonic expansion is used for the angular direction. The new constraint allows for significant savings even on relatively simple solutions as demonstrated on the so-called full sphere benchmark, a specific problem with a very accurately-known solution. The numerical implementation uses a 2D data decomposition which allows it to scale to thousands of cores on present-day high performance computing systems. In addition to validation results, we also present three new whole sphere dynamo solutions that present a relatively simple structure.
Marti, P., & Jackson, A. (2016). A fully spectral methodology for magnetohydrodynamic calculations in a whole sphere. Journal of Computational Physics, 305, 403–422. https://doi.org/10.1016/j.jcp.2015.10.056