Constraint violation in free evolution schemes: comparing BSSNOK with a conformal decomposition of Z4

TL;DR

This paper compares BSSNOK and a conformal Z4 formulation in numerical relativity, showing that the propagating constraints of Z4 improve stability and accuracy, especially with matter, with minimal implementation changes.

Contribution

It introduces and evaluates a conformal Z4 formulation that propagates constraints, demonstrating advantages over BSSNOK in numerical simulations of spacetime.

Findings

Z4 formulation has better constraint handling in matter-rich simulations.

Propagating constraints lead to improved numerical stability.

Implementation of Z4 is a simple modification of BSSNOK.

Abstract

We compare numerical evolutions performed with the BSSNOK formulation and a conformal decomposition of a Z4-like formulation of General Relativity. The important difference between the two formulations is that the Z4 formulation has a propagating Hamiltonian constraint, whereas BSSNOK has a zero-speed characteristic variable in the constraint subsystem. In spherical symmetry we evolve both puncture and neutron star initial data. We demonstrate that the propagating nature of the Z4 constraints leads to results that compare favorably with BSSNOK evolutions, especially when matter is present in the spacetime. From the point of view of implementation the new system is a simple modification of BSSNOK.