Momentum constraint relaxation

TL;DR

This paper introduces a numerical method to control momentum constraint violations in 3D relativistic simulations, improving stability and accuracy in modeling binary neutron star systems.

Contribution

The paper presents a novel momentum constraint relaxation technique based on a vector potential, enhancing the control of constraint violations in relativistic simulations.

Findings

Effectively controls momentum constraint violations in simulations.

Prevents instabilities caused by full enforcement of constraints.

Improves the stability and accuracy of binary neutron star simulations.

Abstract

Full relativistic simulations in three dimensions invariably develop runaway modes that grow exponentially and are accompanied by violations of the Hamiltonian and momentum constraints. Recently, we introduced a numerical method (Hamiltonian relaxation) that greatly reduces the Hamiltonian constraint violation and helps improve the quality of the numerical model. We present here a method that controls the violation of the momentum constraint. The method is based on the addition of a longitudinal component to the traceless extrinsic curvature generated by a vector potential w_i, as outlined by York. The components of w_i are relaxed to solve approximately the momentum constraint equations, pushing slowly the evolution toward the space of solutions of the constraint equations. We test this method with simulations of binary neutron stars in circular orbits and show that effectively controls the growth of the aforementioned violations. We also show that a full numerical enforcement of the constraints, as opposed to the gentle correction of the momentum relaxation scheme, results in the development of instabilities that stop the runs shortly.