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

## TL;DR

This paper introduces a new algorithm for efficiently computing the 3D gravitational potential on multi-patch spherical grids, reducing computational cost and communication overhead while maintaining accuracy.

## Contribution

The proposed method computes the gravitational potential directly across all patches using spherical harmonics, improving efficiency and parallel scalability over previous approaches.

## Key findings

- Achieves the same accuracy as previous methods
- Reduces computational cost significantly
- Improves parallel scaling performance

## 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.