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Geosci. Model Dev., 11, 1497-1515, 2018
https://doi.org/10.5194/gmd-11-1497-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
Development and technical paper
17 Apr 2018
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, P.O. 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 explicit – 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 for some methods. 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., 11, 1497-1515, https://doi.org/10.5194/gmd-11-1497-2018, 2018.

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As the computational power of supercomputing 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 splittings, methods, and solvers on two test cases. Based on these results, we identify the most accurate and efficient approaches for consideration in production models.
As the computational power of supercomputing systems increases, and models for simulating the...
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