Constraint Propagation of $C^2$-adjusted Formulation - Another Recipe for Robust ADM Evolution System

TL;DR

This paper introduces a new constraint adjustment method for the ADM evolution system, enhancing numerical stability in Einstein equation simulations by damping constraint violations, demonstrated through Gowdy-wave tests.

Contribution

It proposes a $C^2$-adjusted formulation for the ADM system, providing a pre-determined effective signature for constraint damping and improved robustness over standard methods.

Findings

Demonstrated eigenvalues indicating constraint damping effectiveness.

Numerical tests show increased robustness against constraint violations.

Outperforms standard ADM in Gowdy-wave simulations.

Abstract

With a purpose of constructing a robust evolution system against numerical instability for integrating the Einstein equations, we propose a new formulation by adjusting the ADM evolution equations with constraints. We apply an adjusting method proposed by Fiske (2004) which uses the norm of the constraints, C2. One of the advantages of this method is that the effective signature of adjusted terms (Lagrange multipliers) for constraint-damping evolution is pre-determined. We demonstrate this fact by showing the eigenvalues of constraint propagation equations. We also perform numerical tests of this adjusted evolution system using polarized Gowdy-wave propagation, which show robust evolutions against the violation of the constraints than that of the standard ADM formulation.