  We consider a model of a dilute Bose-Einstein condensed gas at finite
temperatures, where the condensate coexists in a trap with a cloud of thermal
excitations. Within the ZGN formalism, the dynamics of the condensate is
described by a generalized Gross-Pitaevskii equation, while the thermal cloud
is represented by a semiclassical kinetic equation. Our numerical approach
simulates the kinetic equation using a cloud of representative test particles,
while collisions are treated by means of a Monte Carlo sampling technique. A
full 3D split-operator Fast Fourier Transform method is used to evolve the
condensate wavefunction. We give details regarding the numerical methods used
and discuss simulations carried out to test the accuracy of the numerics. We
use this scheme to simulate the monopole mode in a spherical trap. The
dynamical coupling between the condensate and thermal cloud is responsible for
frequency shifts and damping of the condensate collective mode. We compare our
results to previous theoretical approaches, not only to confirm the reliability
of our numerical scheme, but also to check the validity of approximations which
have been used in the past.
