AMR, stability and higher accuracy

TL;DR

This paper advocates for a higher order accurate adaptive mesh refinement scheme in numerical relativity, combining adaptive gridding with higher order schemes to improve accuracy and efficiency.

Contribution

It introduces a novel stable method for implementing higher order adaptive mesh refinement, addressing boundary stability issues.

Findings

Standard high order adaptive schemes exhibit unstable modes.

A new boundary treatment stabilizes high order adaptive schemes.

The proposed method shows promise for accurate numerical relativity simulations.

Abstract

Efforts to achieve better accuracy in numerical relativity have so far focused either on implementing second order accurate adaptive mesh refinement or on defining higher order accurate differences and update schemes. Here, we argue for the combination, that is a higher order accurate adaptive scheme. This combines the power that adaptive gridding techniques provide to resolve fine scales (in addition to a more efficient use of resources) together with the higher accuracy furnished by higher order schemes when the solution is adequately resolved. To define a convenient higher order adaptive mesh refinement scheme, we discuss a few different modifications of the standard, second order accurate approach of Berger and Oliger. Applying each of these methods to a simple model problem, we find these options have unstable modes. However, a novel approach to dealing with the grid boundaries introduced by the adaptivity appears stable and quite promising for the use of high order operators within an adaptive framework.