  We study Landau damping in dilute Bose-Einstein condensed gases in both
spherical and prolate ellipsoidal harmonic traps. We solve the Bogoliubov
equations for the mode spectrum in both of these cases, and calculate the
damping by summing over transitions between excited quasiparticle states. The
results for the spherical case are compared to those obtained in the
Hartree-Fock approximation, where the excitations take on a single-particle
character, and excellent agreement between the two approaches is found. We have
also taken the semiclassical limit of the Hartree-Fock approximation and obtain
a novel expression for the Landau damping rate involving the time dependent
self-diffusion function of the thermal cloud. As a final approach, we study the
decay of a condensate mode by making use of dynamical simulations in which both
the condensate and thermal cloud are evolved explicitly as a function of time.
A detailed comparison of all these methods over a wide range of sample sizes
and trap geometries is presented.
