Implicit-Explicit (IMEX) Runge-Kutta Methods for Non-Hydrostatic Atmospheric Models

TitleImplicit-Explicit (IMEX) Runge-Kutta Methods for Non-Hydrostatic Atmospheric Models
Publication TypeJournal Article
Year of Publication2018
JournalGeoscientific Model Development
Date Published02/2018
Abstract / Summary

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 implicit-vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the 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 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.

URLhttps://doi.org/10.5194/gmd-2017-285
DOI10.5194/gmd-2017-285
Journal: Geoscientific Model Development
Year of Publication: 2018
Date Published: 02/2018

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 implicit-vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the 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 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.

DOI: https://doi.org/10.5194/gmd-2017-285
Citation:
Gardner, DJ, JE Guerra, FP Hamon, DR Reynolds, PA Ullrich, and CS Woodward.  2018.  "Implicit-Explicit (IMEX) Runge-Kutta Methods for Non-Hydrostatic Atmospheric Models."  Geoscientific Model Development.  https://doi.org/https://doi.org/10.5194/gmd-2017-285.