Gravitational Self-force Errors of Poisson Solvers on Adaptively Refined Meshes

TL;DR

This paper investigates self-force errors in gravitational simulations on adaptive meshes, demonstrating their causes, effects, and methods for correction to improve physical accuracy in numerical models.

Contribution

It provides an analytical expression for self-force errors and introduces correction techniques to mitigate these errors in adaptive mesh gravitational simulations.

Findings

Self-force errors arise from nonuniform mesh refinement.

Errors can be corrected to arbitrary order with boundary correction terms.

Uncorrected errors lead to unphysical energy and momentum changes.

Abstract

An error in the gravitational force that the source of gravity induces on itself (a self-force error) violates both the conservation of linear momentum and the conservation of energy. If such errors are present in a self-gravitating system and are not sufficiently random to average out, the obtained numerical solution will become progressively more unphysical with time: the system will acquire or lose momentum and energy due to numerical effects. In this paper, we demonstrate how self-force errors can arise in the case where self-gravity is solved on an adaptively refined mesh when the refinement is nonuniform. We provide the analytical expression for the self-force error and numerical examples that demonstrate such self-force errors in idealized settings. We also show how these errors can be corrected to an arbitrary order by straightforward addition of correction terms at the refinement boundaries.