  We report on a series of tests of Newtonian Lagrangian perturbation schemes
using N--body simulations for various power--spectra with scale--independent
indices in the range $-3$ to $+1$. The models have been evolved deeply into the
non--linear regime of structure formation in order to probe the dynamical and
statistical performance of the Lagrangian perturbation schemes (whose
first--order solution contains as a subset the celebrated
``Zel'dovich--approximation'' (hereafter ZA). These tests reveal properties of
the approximations at stages beyond the obvious validity of perturbation
theory. Recently, another series of tests of different analytical and
semi--numerical approximations for large--scale structure was conducted with
the result that ZA displays the best dynamical performance in comparison with
the N--body simulations, if the initial data were smoothed before evolving the
model, i.e., a truncated form of ZA (TZA). We show in this Letter that the
excellent performance of TZA can be further improved by going to second order
in the Lagrangian perturbation approach. The truncated second--order Lagrangian
scheme provides a useful improvement over TZA especially for negative power
indices, which suggests it will be very useful for modelling standard scenarios
such as ``Cold--'', ``Hot--'' and ``Mixed--Dark--Matter''.
