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Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 4.0 License.
Development and technical paper
09 Nov 2017
Review status
This discussion paper is a preprint. A revision of this manuscript was accepted for the journal Geoscientific Model Development (GMD) and is expected to appear here in due course.
Implicit-explicit (IMEX) Runge-Kutta methods for non-hydrostatic atmospheric models
David J. Gardner1, Jorge E. Guerra2, François P. Hamon3, Daniel R. Reynolds4, Paul A. Ullrich2, and Carol S. Woodward1 1Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94550, USA
2Department of Land, Air and Water Resources, University of California, Davis, One Shields Ave., Davis, CA 95616, USA
3Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley, CA 94720
4Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75257, USA
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 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.

Citation: Gardner, D. J., Guerra, J. E., Hamon, F. P., Reynolds, D. R., Ullrich, P. A., and Woodward, C. S.: Implicit-explicit (IMEX) Runge-Kutta methods for non-hydrostatic atmospheric models, Geosci. Model Dev. Discuss.,, in review, 2017.
David J. Gardner et al.
David J. Gardner et al.


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Short summary
The as computational power of super computing systems increases and models for simulating the fluid flow of the Earth's atmosphere operate at higher resolutions, new approaches for advancing these models in time will be necessary. In order to produce the best possible result in the least amount of time we evaluate a number of methods and related options in two test cases. Based on these results we identify the most accurate and efficient approaches for consideration in production models.
The as computational power of super computing systems increases and models for simulating the...