By using a full-implicit-continuous-Eulerian (FICE) scheme and taking a set of basic atmospheric motion equations in spherical coordinates as governing equations, a fully nonlinear numerical model for the dynamics of the middle and upper atmosphere is established to numerically study the nonlinear global propagations and amplitude growths of large-scale gravity wave packets. The simulation results show that the newly established model can successfully exhibit the essential characteristics of the global propagations of gravity waves: wave energies propagate upward along the ray paths derived from the linear gravity wave theory, and wave disturbance amplitudes increase with the increasing heights. During the upward propagation of gravity wave packets, there occurs evident enhancement of background wind along the propagation direction. Moreover, our simulation demonstrates that Earth rotation has little influences on the wave energy propagations and amplitude growths. However, the primary nonlinear curvature terms ($frac{uv}{r}$ tan$theta$ and - $frac{u^{2}}{r}$ tan$theta$) play an important role in the growths of gravity wave amplitudes, which causes a evident latitudinal dependence of wave amplitude growth. In the Northern Hemisphere, with the same spatial scales and initial zonal disturbance amplitudes, the waves propagating in lower-latitude regions have larger disturbance amplitudes.