  We study the truncated Wigner method (TWM) applied to a weakly interacting
Bose condensed gas perturbed away from thermal equilibrium. The idea of the
method is to generate an ensemble of classical fields which samples the Wigner
function of the initial thermal density operator, and to evolve each field with
the Gross-Pitaevskii equation (GPE). In the first part of the paper we improve
the sampling technique over our previous work and we test its accuracy against
the exactly solvable model of the ideal gas. In the second part of the paper we
investigate the conditions of validity of the TWM. For short evolution times
the time-dependent Bogoliubov approximation is valid for almost pure
condensates. The requirement that the TWM reproduces the Bogoliubov prediction
leads to the constraint that the number of field modes must be smaller than the
number of particles.For longer times the nonlinear dynamics of the noncondensed
modes plays an important role. To demonstrate this we analyse the case of a 3D
spatially homogeneous Bose condensed gas and we test how well TWM reproduces
Beliaev-Landau damping.We have identified the mechanism which limits the
validity of the TWM: the initial classical fields, driven by the time-dependent
GPE, thermalise to a classical field distribution at a temperature T_class
larger than the initial temperature T. When T_class significantly exceeds T a
spurious damping is observed in the simulation. This leads to the second
condition for the TWM, T_class-T<<T, which requires that the maximum energy of
the Bogoliubov modes does not exceed a few k_B T.
