A Generalized Solution Method for Parallelized Computation of the Three-dimensional Gravitational Potential on a Multi-patch grid in Spherical Geometry

Abstract: We present a generalized algorithm based on a spherical harmonics expansion method for efficient computation of the three-dimensional gravitational potential on a multi-patch grid in spherical geometry. Instead of solving for the gravitational potential by superposition of separate contributions from the mass density distribution on individual grid patch our new algorithm computes directly the gravitational potential due to contributions from all grid patches in one computation step, thereby reducing the computational cost of the gravity solver. This is possible by considering a set of angular weights which are derived from rotations of spherical harmonics functions defined in a global coordinate system that is common for all grid patches. Additionally, our algorithm minimizes data communication between parallel compute tasks by eliminating its proportionality to the number of subdomains in the grid configuration, making it suitable for parallelized computation on a multi-patch grid configuration with any number of subdomains. Test calculations of the gravitational potential of a tri-axial ellipsoidal body with constant mass density on the Yin-Yang two-patch overset grid demonstrate that our method delivers the same level of accuracy as a previous method developed for the Yin-Yang grid, while offering improved computation efficiency and parallel scaling behaviour.