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

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 us 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 a Schwarzschild space-time. In those examples, we find advantages of the strongly hyperbolic formulations over the Arnowitt-Deser-Misner (ADM) system presented in [28]. Furthermore, we observe stabilizing effects of the partially constrained evolution in Schwarzschild spacetime as long as the momentum constraints are enforced.