  The Lagrangian perturbation theory on Friedmann-Lemaitre cosmologies is
compared with numerical simulations (tree-, adaptive P$^3$M- and PM codes). In
previous work we have probed the large-scale performance of the Lagrangian
perturbation solutions up to the third order by studying their cross-
correlations with N-body simulations for various power spectra (Buchert etal
1994, Melott etal 1995, Weiss etal 1996). Thereby, spatial optimization
techniques were applied by (high-frequency-)filtering of the initial power
spectra. In this work the novel method of temporal optimization [Shifted-Time-
Approximation (STA) and Frozen-Time-Approximation (FTA)] is investigated and
used. The method is designed to compensate the native property of Lagrangian
perturbation solutions to delay the collapse of structures. The method can be
treated analytically. Applying the STA and FTA prescriptions a significant
improvement of the performance of Lagrangian perturbation schemes (as measured
by cross-correlation statistics) is observed. Using this tool we investigate a
local study of special clustering models of dark matter as candidates for
typical elements of the large-scale structure in the Universe, and so also
focus on the performance of the perturbation solutions on smaller scales at
high-spatial resolution. The models analyzed were presented in (Buchert etal
1996) and allow studying typical features of the clustering process in the
weakly non-linear regime. The spatial and temporal limits of applicability of
the solutions at second and third order are determined and compared with the
first-order solution, which is equivalent to the ``Zel'dovich approximation''
(Zel'dovich 1970, 1973) for the type of initial data analyzed.
