  I report on controlled comparison of gravitational approximation schemes
linear/lognormal/adhesion/frozen-flow/Zel'dovich(ZA) and ZA's second--order
generalization. In the last two cases we also created new versions of the
approximation by truncation, i.e., by finding an optimum smoothing window (see
text) for the initial conditions. The Zel'dovich approximation, with optimized
initial smoothing, worked extremely well. Its second-order generalization was
slightly better. The success of our best-choice was a result of the treatment
of the phases of nonlinear Fourier components. The adhesion approximation
produced the most accurate nonlinear power spectrum and density distribution,
but its phase errors suggest mass condensations were moved somewhat
incorrectly. Due to its better reproduction of the mass density distribution
function and power spectrum, adhesion might be preferred for some uses. We
recommend either n-body simulations or our modified versions of ZA, depending
on the purpose. Modified ZA can rapidly generate large numbers of realizations
of model universes with good accuracy down to galaxy group (or smaller) mass
scales.
