Cure for unstable numerical evolutions of single black holes: Adjusting the standard ADM equations in the spherically symmetric case

Numerical codes based on a direct implementation of the standard Arnowitt-Deser-Misner (ADM) formulation of Einstein’s equations have generally failed to provide long-term stable and convergent evolutions of black hole space-times when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding terms involving the constraints to the evolution equations, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi-well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives.