Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration

TL;DR

This paper introduces strongly hyperbolic Hamiltonian formulations of general relativity and evaluates their numerical stability and accuracy using symplectic integrators in simulations of Minkowski and Schwarzschild space-times.

Contribution

It presents new strongly hyperbolic Hamiltonian formulations with hyperbolic drivers and demonstrates their advantages over traditional ADM systems in numerical relativity.

Findings

Strong hyperbolic formulations outperform ADM in stability.

Partially constrained evolution stabilizes Schwarzschild simulations.

Enforcing momentum constraints improves numerical accuracy.

Abstract

We consider two strongly hyperbolic Hamiltonian formulations of general relativity and their numerical integration with a free and a partially constrained symplectic integrator. In those formulations we use hyperbolic drivers for the shift and in one case also for the densitized lapse. A system where the densitized lapse is an external field allows to enforce the momentum constraints in a holonomically constrained Hamiltonian system and to turn the Hamilton constraint function from a weak to a strong invariant. These schemes are tested in a perturbed Minkowski and the Schwarzschild space-time. In those examples we find advantages of the strongly hyperbolic formulations over the ADM system presented in [arXiv:0807.0734]. Furthermore we observe stabilizing effects of the partially constrained evolution in Schwarzschild space-time as long as the momentum constraints are enforced.