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 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 multi-step methods (e.g., Adams methods and backward difference formulas) and multi-stage 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, 1D 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. SIAM J. Sci. Comp. \201in press 2013\202