Optimized Lagrangian Approximations for Modelling Large--Scale Structure   at Non--Linear Stages
  Approximations to the exact solutions for gravitational instability in the expanding Universe are extremely useful for understanding the evolution of large--scale structure. 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 contribution 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''. 
T. Buchert, A.L. Melott, A.G. Weiss