Implicit-explicit (IMEX) Runge-Kutta methods for non-hydrostatic atmospheric models

Abstract

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work we investigate various implicit-explicit (IMEX) additive Runge-Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally implicitvertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case the 5 impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a 10 globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy. Overall, the third order ARS343 and ARK324 methods performed the best, followed by the second order ARS232 and ARK232 methods.

DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models

“…ACME-A and Tempest both use the ARS(2,3,2) scheme described in Ascher et al (1997), with all horizontal and vertical advection terms treated explicitly and the remaining vertical terms, associated with sound wave propagation, treated implicitly. A number of different ARK schemes have been compared and contrasted in this framework, with significant implications for model performance and stability (Gardner et al, 2017).…”

Evaluation of Implicit‐Explicit Additive Runge‐Kutta Integrators for the HOMME‐NH Dynamical Core

“…The two dynamical cores do, however, use different model formulations for the atmosphere. Unlike the formulation studied in Gardner et al (), there are only two forcing terms responsible for vertical acoustic wave propagation in the HOMME‐NH formulation. As such, this work considers a single splitting that treats only those two terms implicitly.…”

“…However, these equations either cannot be employed on all scales or require global communication at each time step (i.e., Davies et al, ; Klein et al, ). A second and commonly used alternative among operational nonhydrostatic models (Ullrich et al, ) is the use of implicit‐explicit additive Runge‐Kutta (ARK IMEX) methods (Ascher et al, ; Gardner et al, ; Ullrich & Jablonowski, ; Weller et al, ). These methods work by distinguishing both “slow” and “fast” terms in the spatially discrete fluid equations, the latter set including vertically propagating sound waves.…”

“…The ARK IMEX methods treat the “fast” terms implicitly while retaining an explicit treatment of the “slow” terms, with coefficients and ordering devised in such a manner as to ensure stability and accuracy. A vast library of ARK IMEX methods are now available throughout the published literature, but only a few studies have assessed these methods in the context of nonhydrostatic global atmospheric models (Gardner et al, ; Giraldo et al, ). It is thus the goal of this paper to outline a number of metrics that can be used in performing this assessment and apply these metrics in the context of a new nonhydrostatic dynamical core.…”

“…The dynamical core studied in Gardner et al () is similar to HOMME‐NH in that both use the spectral element method and, therefore, have a spatial operator with purely imaginary eigenvalues. Furthermore, both dynamical cores partition the computational grid such that every vertical column is placed on only one computational node.…”