Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)

We derive an implicit-explicit (IMEX) formalism for the three-dimensional (3D) Euler equations that allow a unified representation of various nonhydrostatic flow regimes, including cloud resolving and mesoscale (flow in a 3D Cartesian domain) as well as global regimes (flow in spherical geometries). This general IMEX formalism admits numerous types of methods including single-stage multistep methods (e.g., Adams methods and backward difference formulas) and multistage single-step methods (e.g., additive Runge--Kutta methods). The significance of this result is that it allows a numerical model to reuse the same machinery for all classes of time-integration methods described in this work. We also derive two classes of IMEX methods, one-dimensional and 3D, and show that they achieve their expected theoretical rates of convergence regardless of the geometry (e.g., 3D box or sphere) and introduce a new second-order IMEX Runge--Kutta method that performs better than the other second-order methods considered. We then compare all the IMEX methods in terms of accuracy and efficiency for two types of geophysical fluid dynamics problems: buoyant convection and inertia-gravity waves. These results show that the high-order time-integration methods yield better efficiency particularly when high levels of accuracy are desired.