A well-balanced Discontinuous Galerkin scheme for the Einstein-Euler equations

I show numerical results obtained with the first-order Z4 formulation of the Einstein--Euler equations. I use a Discontinuous Galerkin (DG) scheme in the ADER version, combined with a subcell limiter to be activated in the presence of strong gradients. In addition, a new pragmatic version of well balancing is proposed, which amounts to subtracting the equilibrium solution from the evolved one all along the simulation. All fundamental tests of numerical relativity are successfully reproduced, reaching three remarkable achievements: (i) long term simulations of stationary, including extreme Kerr black holes, which, acted upon by perturbations, recover the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (rho=p=0) up t=1000 M, with no need to introduce any low density atmosphere; (iii) the head on collision of two punctures black holes is successfully evolved, that was previously considered un--tractable within the Z4 formalism.