  We present a method for solving the stochastic projected Gross-Pitaevskii
equation (SPGPE) for a three-dimensional Bose gas in a harmonic-oscillator
trapping potential. The SPGPE contains the challenge of both accurately
evolving all modes in the low energy classical region of the system, and
evaluating terms from the number-conserving scattering reservoir process. We
give an accurate and efficient procedure for evaluating the scattering terms
using a Hermite-polynomial based spectral-Galerkin representation, which allows
us to precisely implement the low energy mode restriction. Stochastic
integration is performed using the weak semi-implicit Euler method. We
extensively characterize the accuracy of our method, finding a faster than
expected rate of stochastic convergence. Physical consistency of the algorithm
is demonstrated by considering thermalization of initially random states.
