Constraint Damping in First-Order Evolution Systems for Numerical Relativity

TL;DR

This paper introduces a new formulation for Einstein's equations in numerical relativity that actively dampens constraint violations, ensuring more stable and accurate simulations of black holes.

Contribution

It generalizes the KST first-order system with modified evolution equations that suppress constraint violations while preserving hyperbolicity.

Findings

Constraint violations decay exponentially with proper parameter choices

The new system maintains hyperbolicity of evolution equations

Numerical tests on Schwarzschild black holes confirm effectiveness

Abstract

A new constraint suppressing formulation of the Einstein evolution equations is presented, generalizing the five-parameter first-order system due to Kidder, Scheel and Teukolsky (KST). The auxiliary fields, introduced to make the KST system first-order, are given modified evolution equations designed to drive constraint violations toward zero. The algebraic structure of the new system is investigated, showing that the modifications preserve the hyperbolicity of the fundamental and constraint evolution equations. The evolution of the constraints for pertubations of flat spacetime is completely analyzed, and all finite-wavelength constraint modes are shown to decay exponentially when certain adjustable parameters satisfy appropriate inequalities. Numerical simulations of a single Schwarzschild black hole are presented, demonstrating the effectiveness of the new constraint-damping modifications.