Computing the gravitational potential on nested meshes using the convolution method

TL;DR

This paper introduces an extension of the convolution method for fast, accurate computation of gravitational potential on nested meshes, outperforming iterative methods in speed, especially with GPU acceleration, and suitable for high-contrast astrophysical objects.

Contribution

The paper develops a convolution-based approach for gravitational potential calculation on nested grids, demonstrating significant speed improvements and comparable accuracy to iterative methods.

Findings

Convolution method achieves second-order convergence overall.

Outperforms iterative methods by factors of 10-200 in runtime with GPU use.

Suitable for high-contrast density configurations in astrophysics.

Abstract

Aims. Our aim is to derive a fast and accurate method for computing the gravitational potential of astrophysical objects with high contrasts in density, for which nested or adaptive meshes are required. Methods. We present an extension of the convolution method for computing the gravitational potential to the nested Cartesian grids. The method makes use of the convolution theorem to compute the gravitational potential using its integral form. Results. A comparison of our method with the iterative outside-in conjugate gradient and generalized minimal residual methods for solving the Poisson equation using nonspherically symmetric density configurations has shown a comparable performance in terms of the errors relative to the analytic solutions. However, the convolution method is characterized by several advantages and outperforms the considered iterative methods by factors 10--200 in terms of the runtime, especially when graphics processor units are utilized. The convolution method also shows an overall second-order convergence, except for the errors at the grid interfaces where the convergence is linear. Conclusions. High computational speed and ease in implementation can make the convolution method a preferred choice when using a large number of nested grids. The convolution method, however, becomes more computationally costly if the dipole moments of tightly spaced gravitating objects are to be considered at coarser grids.