  A 10 MeV/c $\mu^+$ beam was stopped in helium gas of a few mbar in a magnetic
field of 5 T. The muon 'swarm' has been efficiently compressed from a length of
16 cm down to a few mm along the magnetic field axis (longitudinal compression)
using electrostatic fields. The simulation reproduces the low energy
interactions of slow muons in helium gas. Phase space compression occurs on the
order of microseconds, compatible with the muon lifetime of 2 $\mu$s. This
paves the way for preparation of a high quality muon beam.
