The DOE Energy Exascale Earth System Model (E3SM) project is developing a new non-hydrostatic dynamical core for modeling atmospheric fluid dynamics at high resolution. The goal of this work was to understand the likely best performing numerical time integrators, splittings, and solver approaches for these systems. Non-hydrostatic motion includes fast waves in the vertical direction which strongly reduce the stable time step for an overall explicit integration approach. Implicit/explicit (IMEX) methods allow for treatment of part of the system with an explicit method and the rest of the system with an implicit method. We have compared many IMEX splittings of the non-hydrostatic dynamics system where the fast vertical waves are handled implicitly and the horizontal dynamics are handled explicitly as well as splittings with some implicit horizontal dynamics. The work used the ARKode time integration package from the SUNDIALS suite to allow for testing over 20 different IMEX methods with different splittings of model equations between explicit and implicit approaches as well as different approaches for solving the implicit systems. The work was done in the Tempest non-hydrostatic dynamical core developed at UC Davis. Methods, splittings, and solvers were evaluated based on the largest time step that could be taken while maintaining a limited variation in solution. The limit of variation was taken from the variation in solution resulting from running a fully explicit method with randomly perturbed initial conditions.  The best performing methods are currently being tested in the new HOMME non-hydrostatic dynamical core in E3SM.