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We consider a model of a dilute Bose-Einstein condensed gas at finitetemperatures, where the condensate coexists in a trap with a cloud of thermalexcitations. Within the ZGN formalism, the dynamics of the condensate isdescribed by a generalized Gross-Pitaevskii equation, while the thermal cloudis represented by a semiclassical kinetic equation. Our numerical approachsimulates the kinetic equation using a cloud of representative test particles,while collisions are treated by means of a Monte Carlo sampling technique. Afull 3D split-operator Fast Fourier Transform method is used to evolve thecondensate wavefunction. We give details regarding the numerical methods usedand discuss simulations carried out to test the accuracy of the numerics. Weuse this scheme to simulate the monopole mode in a spherical trap. Thedynamical coupling between the condensate and thermal cloud is responsible forfrequency shifts and damping of the condensate collective mode. We compare ourresults to previous theoretical approaches, not only to confirm the reliabilityof our numerical scheme, but also to check the validity of approximations whichhave been used in the past.Comment: 33 pages, 8 figure

Modeling Bose-Einstein condensed gases at finite temperatures withN-body simulations