The classical theory of acoustic streaming in a standing wave resonator predicts a parabolic mean velocity profile. A recent analysis by Menguy and Gilbert [Acustica, 86, pp 249-259, 2000] has suggested that a parabolic mean velocity profile is obtained only for a low acoustic pressure amplitude. For higher amplitudes, the streaming velocity deviates significantly from this distribution. Several recent experiments are also in general agreement with Menguy and Gilbert's argument. In the present paper, time-accurate high-order numerical simulations of the full Navier-Stokes equations are performed to study high pressure amplitude standing wave streaming. Both streaming inside the boundary layer and in the resonator core region are considered. The results show that the streaming velocity and the mean temperature variation along the resonator depend significantly on the resonator wall boundary condition. An adiabatic boundary condition results in a flatter streaming velocity profile: whereas, an isothermal boundary condition provides a streaming velocity profile that is closer to a parabolic distribution, even for high amplitude oscillations. The simulation is also used to investigate the time evolution of the mean velocity. It is observed that the time scale associated with the streaming development inside the boundary layer is significantly shorter than that of the outer streaming.