  In this work we aim at developing a new class of high order accurate
well-balanced
  finite difference (FD) Weighted Essentially Non-Oscillatory (WENO) methods
for numerical general relativity,
  which can be applied to any first--order reduction of the Einstein field
equations, even if non--conservative terms are present.
  We choose the first--order non--conservative Z4 formulation of the Einstein
equations, which
  has a built--in cleaning procedure that accounts for the Einstein constraints
and that
  has already shown its ability in keeping stationary solutions stable over
long timescales.
  Upon the introduction of auxiliary variables, the vacuum Einstein equations
in first order form constitute a ...
