This study presents a numerical investigation of acoustic streaming motion (of the Rayleigh type) in a compressible gas inside two-dimensional rectangular enclosures. To numerically study the effects of the sound field intensity on the formation process of streaming structures, we propose to discretize the fully compressible form of the two-dimensional Navier-Stokes equations using a high-order (formally greater or equal to fourth-order) accurate numerical scheme in both space and time. The proposed numerical solver utilizes high-order compact schemes along each spatial dimension combined with a filtering procedure when it is necessary. Acoustic standing waves are excited inside the enclosures and the resulting acoustic streaming patterns are investigated for low and high-intensity waves, in both linear and nonlinear regimes following closely the work of Daru et al. (2013). An extended investigation indicates that without the incorporation of an appropriate filter, the application of the high-order compact schemes in case of fast streaming results in spurious oscillations which inhibit their applicability. Following the recent relevant literature, the numerical simulations performed demonstrate the ability of the proposed numerical approach to reproduce efficiently and robustly the transitions from regular acoustic streaming to irregular streaming and the relevant phenomena confirming results that have been previously presented.