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Excision techniques are used in order to deal with black holes in numericalsimulations of Einstein equations and consist in removing a topological spherecontaining the physical singularity from the numerical domain, applying insteadappropriate boundary conditions at the excised surface. In this work we presentrecent developments of this technique in the case of constrained formulationsof Einstein equations and for spherically symmetric spacetimes. We present anew set of boundary conditions to apply to the elliptic system in thefully-constrained formalism of Bonazzola et al. (2004), at an arbitrarycoordinate sphere inside the apparent horizon. Analytical properties of thissystem of boundary conditions are studied and, under some assumptions, anexponential convergence toward the stationary solution is exhibited for thevacuum spacetime. This is verified in numerical examples, together with theapplicability in the case of the accretion of a scalar field onto aSchwarzschild black hole. We also present the successful use of the excisiontechnique in the collapse of a neutron star to a black hole, when excision isswitched on during the simulation, after the formation of the apparent horizon.This allows the accretion of matter remaining outside the excision surface andfor the stable long-term evolution of the newly formed black hole.

An excision scheme for black holes in constrained evolution formulations: spherically symmetric case