  We present a new fast and efficient approach to model structure formation
with Augmented Lagrangian Perturbation Theory (ALPT). Our method is based on
splitting the displacement field into a long and a short-range component. The
long-range component is computed by second order LPT (2LPT). This approximation
contains a tidal nonlocal and nonlinear term. Unfortunately, 2LPT fails on
small scales due to severe shell crossing and a crude quadratic behaviour in
the low density regime. The spherical collapse (SC) approximation has been
recently reported to correct for both effects by adding an ideal collapse
truncation. However, this approach fails to reproduce the structures on large
scales where it is significantly less correlated with the N-body result than
2LPT or linear LPT (the Zeldovich approximation). We propose to combine both
approximations using for the short-range displacement field the SC solution. A
Gaussian filter with a smoothing radius r_S is used to separate between both
regimes. We use the result of 25 dark matter only N-body simulations to
benchmark at z=0 the different approximations: 1st, 2nd, 3rd order LPT, SC and
our novel combined ALPT model. This comparison demonstrates that our method
improves previous approximations at all scales showing ~25% and ~75% higher
correlation than 2LPT with the N-body solution at k = 1 and 2 h Mpc^-1,
respectively. We conduct a parameter study to determine the optimal range of
smoothing radii and find that the maximum correlation is achieved with r_S = 4
- 5 h^-1 Mpc. This structure formation approach could be used for various
purposes, such as setting-up initial conditions for N-body simulations,
generating mock galaxy catalogues, cosmic web analysis or for reconstructions
of the primordial density fluctuations.
