Modelling non-linear evolution using Lagrangian Perturbation Theory
  (LPT) re-expansions
  We present a new method to calculate formation of cosmological structure in
the Newtonian limit. The method is based on Lagrangian perturbation theory plus
two key theoretical extensions. One advance involves identifying and fixing a
previously ignored gauge-like degree of freedom relating quantities calculated
in LPT to those measured by a preferred Friedmann-Robertson-Walker (FRW)
observer. Handling this connection between calculational and observer frames is
physically essential and ensures a momentum conserving description. The second
extension is to systematically re-expand the equations of motion to increase
LPT's radius of convergence to the maximum future time prior to orbit crossing.
The paper implements a complete algorithm and performs extensive "proof of
principle" tests of the new method, including direct comparison to known
solutions, evaluation of conserved quantities and formal convergence studies.
All are satisfactory. We show convergence is exponential in grid size and
Lagrangian order and polynomial in step size. There are three {\it powerful
advantages} afforded by the new technique: (1) it employs a smooth
representation of all fields and the results are not limited by particle
induced shot-noise errors, (2) it permits the numerical error to be controlled
by changing Lagrangian order and/or number of steps allowing, in principle,
arbitrarily small errors to be achieved prior to orbit crossing and (3) it
handles generic cold initial data (any periodic density and velocity fields,
including those with initial rotational components). Together, these properties
make the new technique well-suited to handle quasi-linear scales where analytic
methods and/or numerical simulations fail to provide suitably accurate answers.
