Free and constrained symplectic integrators for numerical general relativity

TL;DR

This paper explores symplectic integrators for numerical general relativity, proposing methods for free and constrained evolution that improve stability and constraint preservation, demonstrated through simplified Einstein equations models.

Contribution

It introduces the use of the Stoermer-Verlet and RATTLE symplectic integrators for free and constrained evolution in numerical relativity, enhancing stability and constraint enforcement.

Findings

Symplectic integrators outperform non-symplectic ones in conserving quantities.

Enforcing momentum constraints stabilizes evolution and suppresses instabilities.

Constrained evolution can turn unstable free evolution into stable simulations.

Abstract

We consider symplectic time integrators in numerical General Relativity and discuss both free and constrained evolution schemes. For free evolution of ADM-like equations we propose the use of the Stoermer-Verlet method, a standard symplectic integrator which here is explicit in the computationally expensive curvature terms. For the constrained evolution we give a formulation of the evolution equations that enforces the momentum constraints in a holonomically constrained Hamiltonian system and turns the Hamilton constraint function from a weak to a strong invariant of the system. This formulation permits the use of the constraint-preserving symplectic RATTLE integrator, a constrained version of the Stoermer-Verlet method. The behavior of the methods is illustrated on two effectively 1+1-dimensional versions of Einstein's equations, that allow to investigate a perturbed Minkowski problem and the Schwarzschild space-time. We compare symplectic and non-symplectic integrators for free evolution, showing very different numerical behavior for nearly-conserved quantities in the perturbed Minkowski problem. Further we compare free and constrained evolution, demonstrating in our examples that enforcing the momentum constraints can turn an unstable free evolution into a stable constrained evolution. This is demonstrated in the stabilization of a perturbed Minkowski problem with Dirac gauge, and in the suppression of the propagation of boundary instabilities into the interior of the domain in Schwarzschild space-time.