  We consider the gravitational collapse of collisionless matter seeded by
three crossed sine waves with various amplitudes, also in the presence of a
linear external tidal field. We explore two theoretical methods that are more
efficient than standard Lagrangian perturbation theory (LPT) for resolving
shell-crossings, the crossing of particle trajectories. One of the methods
completes the truncated LPT series for the displacement field far into the UV
regime, thereby exponentially accelerating its convergence while at the same
time removing pathological behavior of LPT observed in void regions. The other
method exploits normal-form techniques known from catastrophe theory, which
amounts here to replacing the sine-wave initial data by its second-order Taylor
expansion in space at shell-crossing location. This replacement leads to a
speed-up in determining the displacement field by several orders of magnitudes,
while still achieving permille-level accuracy in the prediction of the
shell-crossing time. The two methods can be used independently, but the overall
best performance is achieved when combining them. Lastly, we find accurate
formulas for the nonlinear density and for the triaxial evolution of the fluid
in the fundamental coordinate system, as well as report a newly established
correspondence between perfectly symmetric sine-wave collapse and spherical
collapse.
