  Accurate predictions on non--linear power spectra, at various redshift z,
will be a basic tool to interpret cosmological data from next generation mass
probes, so obtaining key information on Dark Energy nature. This calls for high
precision simulations, covering the whole functional space of w(z) state
equations and taking also into account the admitted ranges of other
cosmological parameters; surely a difficult task. A procedure was however
suggested, able to match the spectra at z=0, up to k~3, hMpc^{-1}, in
cosmologies with an (almost) arbitrary w(z), by making recourse to the results
of N-body simulations with w = const. In this paper we extend such procedure to
high redshift and test our approach through a series of N-body gravitational
simulations of various models, including a model closely fitting WMAP5 and
complementary data. Our approach detects w= const. models, whose spectra meet
the requirement within 1% at z=0 and perform even better at higher redshift,
where they are close to a permil precision. Available Halofit expressions,
extended to (constant) w \neq -1 are unfortunately unsuitable to fit the
spectra of the physical models considered here. Their extension to cover the
desired range should be however feasible, and this will enable us to match
spectra from any DE state equation.
