Adaptive Mesh Refinement for Characteristic Grids

TL;DR

This paper introduces an efficient Berger-Oliger adaptive mesh refinement algorithm for characteristic grids solving wave-like PDEs, improving implementation simplicity and resource management by recursing on null slices instead of individual cells.

Contribution

It presents a novel AMR algorithm that recurses on null slices, simplifying implementation and reducing memory overhead compared to previous cell-based methods.

Findings

The new algorithm is more space-efficient.

It achieves 2nd and 4th order global accuracy.

Code is publicly available under GPL.

Abstract

I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when numerically solving partial differential equations with wave-like solutions, using characteristic (double-null) grids. Such AMR algorithms are naturally recursive, and the best-known past Berger-Oliger characteristic AMR algorithm, that of Pretorius & Lehner (J. Comp. Phys. 198 (2004), 10), recurses on individual "diamond" characteristic grid cells. This leads to the use of fine-grained memory management, with individual grid cells kept in 2-dimensional linked lists at each refinement level. This complicates the implementation and adds overhead in both space and time. Here I describe a Berger-Oliger characteristic AMR algorithm which instead recurses on null \emph{slices}. This algorithm is very similar to the usual Cauchy Berger-Oliger algorithm, and uses relatively coarse-grained memory management, allowing entire null slices to be stored in contiguous arrays in memory. The algorithm is very efficient in both space and time. I describe discretizations yielding both 2nd and 4th order global accuracy. My code implementing the algorithm described here is included in the electronic supplementary materials accompanying this paper, and is freely available to other researchers under the terms of the GNU general public license.