Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems

TL;DR

This paper introduces a modified adaptive mesh refinement algorithm tailored for coupled elliptic-hyperbolic PDE systems, improving simulations in general relativity by handling non-local elliptic solutions efficiently.

Contribution

It presents a novel 'extrapolation and delayed solution' technique integrated into the Berger and Oliger framework for better elliptic-hyperbolic system simulation.

Findings

Effective in axisymmetric gravitational collapse simulations

Improves handling of non-local elliptic solutions

Reduces spurious high-frequency noise

Abstract

We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components ("noise").