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Global non-hydrostatic atmospheric models are becoming increasingly important for studying the climates of planets and exoplanets. However, such models suffer from computational difficulties due to the large aspect ratio between the horizontal and vertical directions. To overcome this problem, we developed a global model using a vertically-implicit-correction (VIC) scheme in which the integration time step is no longer limited by the propagation of acoustic waves in the vertical. We proved that our model, based on the $\rm Athena^{++}$ framework and its extension for planetary atmospheres - SNAP (Simulating Non-hydrostatic Atmosphere on Planets), rigorously conserves mass and energy in finite volume simulations. We found that traditional numerical stabilizers such as hyper-viscosity and divergence damping are not needed when using the VIC scheme, which greatly simplifies the numerical implementation and improves stability. We present simulation results ranging from 1D linear waves to 3D global circulations with and without the VIC scheme. These tests demonstrate that our formulation correctly tracks local turbulent motions, produces Kelvin-Helmholtz instability, and generates a super-rotating jet on hot Jupiters. Employing this VIC scheme improves the computational efficiency of global simulations by more than two orders of magnitude compared to an explicit model and facilitates the capability of simulating a wide range of planetary atmospheres both regionally and globally.