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

Abstract

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 singlestep 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. Introduction.In a previous article [20] we introduced the nonhydrostatic unified model of the atmosphere (NUMA) for use in limited-area modeling (i.e., mesoscale or regional flow), namely, applications in which the flows are in large, three-dimensional (3D) Cartesian domains (imagine flow in a 3D box where the grid resolutions are below 10 km); the emphasis of that paper was on the performance of the model on distributed-memory computers with a large number of processors. In that paper we showed that the explicit RK35 time integrator (also used in this paper) was able to achieve strong linear scaling for processor counts on the order of 10 5 . The emphasis of the present article is on the mathematical framework of the model dynamics (i.e., we are not considering the subgrid-scale parameterization at this point;

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

“…What this means for the IMEX method is that the solution procedure will require iterative (elliptic) solvers which are usually handled via matrix-free approaches since the resulting system of equations will be too large to store in memory-for HEVI methods, the resulting system is quite small and hence direct solvers are appropriate. Note that the HEVI and IMEX methods can be written within the same unified temporal discretization as described in [40,106] where the HEVI scheme is denoted as the 1d-IMEX method, meaning that HEVI is simply an IMEX method that has been partitioned to be implicit in only one of the spatial directions. Further note that whereas HEVI schemes only circumvent the CFL condition related to the vertically-propagating acoustic waves, IMEX methods entirely circumvent the CFL condition related to all acoustic waves.…”

“…Both subcategories, multistage Runge-Kutta and linear multistep methods, can be fullyimplicit (the current time-level is obtained by solving a nonlinear problem that uses information from the current time-step) and fully-explicit (the current time-level is calculated using information coming from the previous time-steps only). Representative classes of time-integration schemes embedded in the GL method consist of implicit multistep methods such as Adams-Moulton (AM) [22] and backward differentiation (BDF) methods [13,20,21], implicit multistage Runge-Kutta schemes such as diagonally (DIRK) and singly-diagonally (SDIRK) implicit Runge-Kutta schemes [3,19,59], explicit multistep methods, such as leapfrog and Adams-Bashforth methods [28,43], explicit Runge-Kutta schemes, such as the fourthorder Runge-Kutta scheme [55] and partitioned methods, such as Implicit-Explicit (IMEX) schemes, whereby the operators are linearized in some fashion with-e.g., two Butcher tableaux, one explicit and one implicit [5,40,106]. While EBTI schemes are widely used in computational fluid dynamics, especially in the engineering sector [18,52], their adoption in the weather and climate communities has been less widespread, with SE schemes [54,88,107] and horizontally-explicit vertically-implicit schemes [8,40,63]-i.e., schemes where the horizontal direction is treated explicitly and the vertical is treated implicitly-becoming more prominent but still confined mainly to research and limited-area models (with very few exceptions-see Table 1).…”

“…The analyses Fig. 3 Illustrations of two commonly used SE integration steps for solving (8): the terms in R g contribute to the model solution using a leapfrog to step from t À Dt to t þ Dt; and b a 3-stage 3rd-order have tended to focus on (single time-step) Runge-Kutta (RK) based approaches [8,23,40,63,100,106], which inherently avoid any problems associated with the computational modes that derive from multi-step methods.…”

“…Expressing the RK-HEVI approach as a double Butcher tableau makes it efficient to explore many alternative combinations of schemes-both through linear analyses [23,40,63] and numerical simulations [23,40,106]. The analyses focus on the accuracy and stability implied for components of simple linear systems (acoustic and gravity waves, and advection); and the performance of numerical implementations for idealised (dry) atmospheric tests, often compared to solutions from a very high-resolution highorder explicit RK method.…”

“…The analyses focus on the accuracy and stability implied for components of simple linear systems (acoustic and gravity waves, and advection); and the performance of numerical implementations for idealised (dry) atmospheric tests, often compared to solutions from a very high-resolution highorder explicit RK method. Substantially different approaches have been recommended: some completely splitting the vertical and horizontal integrations using Strang-type splitting [8,97,100]; others proposing schemes that keep the vertical and horizontal solutions balanced in time, by integrating over the same time-interval, at each predictorstage [40,63,106]. In keeping with the semi-implicit approach (Sect.…”