Toward an efficient second-order method for computing the surface gravitational potential on spherical-polar meshes

TL;DR

This paper compares two methods for efficiently computing the gravitational potential in astrophysical simulations on spherical-polar meshes, demonstrating second-order convergence and superior scalability of James' method.

Contribution

The paper introduces and compares two alternative boundary condition methods for solving the Poisson equation on spherical-polar meshes, highlighting the efficiency of James' method.

Findings

Both methods achieve second-order convergence.

James' method has better algorithmic complexity (~O(n^3)).

James' method scales well with thousands of cores.

Abstract

Astrophysical accretion discs that carry a significant mass compared with their central object are subject to the effect of self-gravity. In the context of circumstellar discs, this can, for instance, cause fragmentation of the disc gas, and -- under suitable conditions -- lead to the direct formation of gas-giant planets. If one wants to study these phenomena, the disc's gravitational potential needs to be obtained by solving the Poisson equation. This requires to specify suitable boundary conditions. In the case of a spherical-polar computational mesh, a standard multipole expansion for obtaining boundary values is not practicable. We hence compare two alternative methods for overcoming this limitation. The first method is based on a known Green's function expansion (termed "CCGF") of the potential, while the second (termed "James' method") uses a surface screening mass approach with a suitable discrete Green's function. We demonstrate second-order convergence for both methods and test the weak scaling behaviour when using thousands of computational cores. Overall, James' method is found superior owing to its favourable algorithmic complexity of $\sim \mathcal{O}(n^3)$ compared with the $\sim\mathcal{O}(n^4)$ scaling of the CCGF method.