  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.
