F. X. Giraldo, J. F. Kelly, E. M. Constantinescu, "Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)," Preprint ANL/MCS-P2083-0512, May 2012. [pdf]
We derive implicit-explicit (IMEX) formulations 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 formulation 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). This is an important result because it allows a numerical model to reuse the same machinery for all 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. We then compare all the IMEX methods in terms of accuracy and efficiency for three types of geophysical fluid dynamics problems: buoyant convection, density current, and a planetary acoustic wave.