  We conduct a thorough study of different forms of horizontally explicit and
vertically implicit (HEVI) time-integration strategies for the compressible
Euler equations on spherical domains typical of nonhydrostatic global
atmospheric applications. We compare the computational time and complexity of
two nonlinear variants (NHEVI-GMRES and NHEVI-LU) and a linear variant (LHEVI).
We report on the performance of these three variants for a number of additive
Runge-Kutta Methods ranging in order of accuracy from second through fifth, and
confirm the expected order of accuracy of the HEVI methods for each
time-integrator. To gauge the maximum usable time-step of each HEVI method, we
run simulations of a nonhydrostatic baroclinic instability for 100 days and
then use this time-step to compare the time-to-solution of each method. The
results show that NHEVI-LU is 2x faster than NHEVI-GMRES, and LHEVI is 5x
faster than NHEVI-LU, for the idealized cases tested. The baroclinic
instability and inertia-gravity wave simulations indicate that the optimal
choice of time-integrator is LHEVI with either second or third order schemes,
as both schemes yield similar time to solution and relative L2 error at their
maximum usable time-steps. In the future, we will report on whether these
results hold for more complex problems using, e.g., real atmospheric data
and/or a higher model top typical of space weather applications.
