Constraint and gauge shocks in one-dimensional numerical relativity

TL;DR

This paper analyzes different blow-up mechanisms in hyperbolic systems like those in general relativity, identifying geometric and ODE-based shocks, including a new type called constraint shocks, and explores how to mitigate them.

Contribution

It introduces the concept of constraint shocks in hyperbolic formulations of relativity and discusses how formulation choices can eliminate or reduce these shocks.

Findings

Identification of geometric and ODE mechanisms for blow-ups.

Discovery of constraint shocks specific to certain formulations.

Strategies to eliminate or mitigate constraint shocks.

Abstract

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known ``gauge shocks'' associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups ``constraint shocks'' and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.