Lowering the Memory Footprint of Fully Spectral Full Sphere Simulations

Abstract:

When modelling planetary cores as a full sphere geometry, great care is required in order to maintain the regularity of the physical fields throughout the volume. One possible approach relies on using a fully spectral code, using spherical harmonics in the angular components, and the so called Jones-Worland polynomials in radius. These are defined as the product of a regularity factor $r^l$ and a Jacobi polynomial $P_n^{\left(\alpha,\beta\right)}(x)$ with quadratic argument, ie $W_n^l \equiv r^l P_n^{\left(-1/2,l-1/2\right)}(2r^2-1)$. Generally, the transforms between physical and spectral space for the Jones-Worland polynomials are implemented using a gaussian quadrature. The accuracy of this approach comes at the price of a large memory footprint, and it is difficult to write an efficient "on-the-fly" formulation. For large scale simulations or applied on HPC systems with accelerators, this becomes a significant limitation. We present a transform algorithm based on the Fast Fourier Transform and recurrence relations of the Jacobi polynomials that tackles these issues. It possesses a similar computational complexity as the gaussian quadrature based computation, but has a smaller memory footprint. Furthermore, it can readily be formulated in an "on-the-fly" form which only requires twice the operation count. By avoiding the explicit evaluation of the basis functions on the grid, our new algorithm also avoids the underflow issues that have to be dealt with at high resolution in a quadrature based implementation.