A new first-order formulation of the Einstein equations: comparison among different high order numerical schemes

Abstract

The solution of the Einstein-Euler equations by the majority of numerical codes is still based on traditional finite difference schemes for the Einstein sector, while it relies on conservative schemes for the matter part. This is due to the fact that the celebrated BSSNOK formulation of the Einstein equations has second order in space derivatives. We present a first-order (in space derivatives) formulation of the BSSNOK Einstein equations that is strongly hyperbolic and it allows for the implementation of a monolithic numerical scheme for its solution. The new formulation is compatible with quite different numerical schemes, such as Central WENO finite differences or Discontinuous Galerkin schemes, that we have analyzed in terms of accuracy and computational performances.