THE PERFORMANCE OF LAGRANGIAN PERTURBATION SCHEMES AT HIGH-RESOLUTION

We present high-spatial resolution studies of the density field as pre dicted by Lagrangian perturbation approximations up to the third order . The first-order approximation is equivalent to the ''Zel'dovich appr oximation'' for the type of initial data analyzed. The study is perfor med for two simple models which allow studying of typical features of the clustering process in the early non-linear regime. We calculate th e initial perturbation potentials as solutions of Poisson equations al gebraically, and automate this calculation for a given initial random density field. The presented models may also be useful for other quest ions addressed to Lagrangian perturbation solutions and for the compar ison of different approximation schemes. In an accompanying paper we i nvestigate a detailed comparison with various N-body integrators using these models (Karakatsanis et al. 1996). Results of the present paper include the following: 1. the collapse is accelerated significantly b y the higher-order corrections confirming previous results by Moutarde et al. (1991); 2. the spatial structure of the density patterns predi cted by the ''Zel'dovich approximation'' differs much from those predi cted by the second- and third-order Lagrangian approximations; 3. seco nd-order effects amount to internal substructures such as ''second gen eration'' -pancakes, -filaments and -clusters, as are also observed in N-body simulations; 4. the third-order effect gives rise to substruct uring of the secondary mass-shells. The hierarchy of shell-crossing si ngularities that form features small high-density clumps at the inters ections of caustics which we interprete as gravitational fragmentation .