  We present a strongly hyperbolic first-order formulation of the Einstein
equations based on the conformal and covariant Z4 system (CCZ4) with
constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this
formulation combines the advantages of a conformal and traceless formulation,
with the suppression of constraint violations given by the damping terms, but
being first order in time and space, it is particularly suited for a
discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order
formulation has been obtained by making careful use of first and second-order
ordering constraints. A proof of strong hyperbolicity is given for a selected
choice of standard gauges via an analytical computation of the entire
eigenstructure of the FO-CCZ4 system. The resulting governing partial
differential equations system is written in non-conservative form and requires
the evolution of 58 unknowns. A key feature of our formulation is that the
first-order CCZ4 system decouples into a set of pure ordinary differential
equations and a reduced hyperbolic system of partial differential equations
that contains only linearly degenerate fields. We implement FO-CCZ4 in a
high-order path-conservative arbitrary-high-order-method-using-derivatives
(ADER)-DG scheme with adaptive mesh refinement and local time-stepping,
supplemented with a third-order ADER-WENO subcell finite-volume limiter in
order to deal with singularities arising with black holes. We validate the
correctness of the formulation through a series of standard tests in vacuum,
performed in one, two and three spatial dimensions, and also present
preliminary results on the evolution of binary black-hole systems. To the best
of our knowledge, these are the first successful three-dimensional simulations
of moving punctures carried out with high-order DG schemes using a first-order
formulation of the Einstein equations.
