Constraint propagation in the family of ADM systems

TL;DR

This paper analyzes the stability of ADM formulations in numerical relativity by examining constraint propagation equations and eigenvalues, proposing methods to enhance robustness against constraint violations.

Contribution

It introduces a framework for analyzing and selecting adjusted ADM systems with favorable eigenvalues to improve stability in numerical relativity simulations.

Findings

Eigenvalues can be controlled by adjusting multipliers of constraint terms.

Stable systems have eigenvalues that are negative or purely imaginary.

The approach encompasses previous proposals like Detweiler's and Frittelli's analyses.

Abstract

The current important issue in numerical relativity is to determine which formulation of the Einstein equations provides us with stable and accurate simulations. Based on our previous work on "asymptotically constrained" systems, we here present constraint propagation equations and their eigenvalues for the Arnowitt-Deser-Misner (ADM) evolution equations with additional constraint terms (adjusted terms) on the right hand side. We conjecture that the system is robust against violation of constraints if the amplification factors (eigenvalues of Fourier-component of the constraint propagation equations) are negative or pure-imaginary. We show such a system can be obtained by choosing multipliers of adjusted terms. Our discussion covers Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also mention the so-called conformal-traceless ADM systems.