  We investigate the performance of Lagrangian perturbation theory up to the
second order for two scenarios of cosmological large-scale structure formation,
SCDM (standard cold dark matter) and BSI (broken scale invariance). The latter
model we study as a representative of COBE-normalized CDM models which fit the
small-scale power of galaxy surveys. In this context we optimize the
performance of the Lagrangian perturbation schemes by smoothing the small-scale
fluctuations in the initial data. The results of the so obtained Lagrangian
mappings are computed for a set of COBE-normalized SCDM and BSI initial data of
different sizes and at different times. We compare these results against those
obtained with a numerical PM-code. We find an excellent performance of the
optimized Lagrangian schemes down to scales close to the correlation length.
This is explained by the counterintuitive fact that nonlinearities in the model
can produce more small-scale power, if initially such power is removed. The
optimization scheme can be expressed in a way which is independent of the type
of fluctuation spectrum and of the size of the simulations.
