Abstract
Effective preconditioning lies at the heart of multiscale flow simulation, including a broad range of geoscientific applications that rely on semi-implicit integrations of the governing PDEs. For such problems, conditioning of the resulting sparse linear operator directly responds to the squared ratio of largest and smallest spatial scales represented in the model. For thin-spherical-shell geometry of the Earth atmosphere the condition number is enormous, upon which implicit preconditioning is imperative to eliminate the stiffness resulting from relatively fine vertical resolution. Furthermore, the anisotropy due to the meridians convergence in standard latitude-longitude discretizations becomes equally detrimental as the horizontal resolution increases to capture nonhydrostatic dynamics. Herein, we discuss a class of effective preconditioners based on the parallel ADI approach. The approach has been implemented in the established high-performance all-scale model EULAG with flexible computational domain distribution, including a 3D processor array. The efficacy of the approach is demonstrated in the context of an archetypal simulation of global weather.
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Notes
- 1.
For comprehensive list of EULAG publications see model webpage at http://www2.mmm.ucar.edu/eulag/.
- 2.
This second step can be iterated when nonlinear terms resulting from, e.g., metric forces are present; cf. [19] for illustrative examples.
- 3.
MPDATA (for multidimensional positive definite advection transport algorithm) is a class of nonoscillatory forward-in-time flow solvers, widely documented in the literature; for a recent overview see [20] and references therein.
- 4.
- 5.
For the majority of geophysical applications, the main performance bottleneck is the memory bandwidth and access time, and not the main memory size. The latter is usually much larger than needed, as large number of timesteps and good scalability lead to routine use of hundreds of computing cores and tens of supercomputer nodes.
- 6.
For a discussion of the differences in various soundproof and compressible solutions see [20] and references therein.
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Acknowledgements
This work was supported by “Towards peta-scale numerical weather prediction for Europe” project realized within the “HOMING PLUS” programme of Foundation for Polish Science, co-financed from European Union, Regional Development Fund. Selected code optimizations were supported by the Polish National Science Center (NCN) under the Grant no.: 2011/03/B/ST6/03500. Piotr K. Smolarkiewicz is supported by funding received from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2012/ERC Grant agreement no. 320375). This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID d25 and by the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) University of Warsaw under grant no. G49-15.
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Piotrowski, Z.P., Matejczyk, B., Marcinkowski, L., Smolarkiewicz, P.K. (2016). Parallel ADI Preconditioners for All-Scale Atmospheric Models. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds) Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science(), vol 9574. Springer, Cham. https://doi.org/10.1007/978-3-319-32152-3_56
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