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A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

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Abstract

Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods. Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287–301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.

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Notes

  1. www.top500.org.

  2. Alternatively, the basis functions can be constructed using Eq. (38).

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Acknowledgments

The authors would like to thank Prof. Eugenio Oñate (U. Politècnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.). Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSC-CNS) and Universitat Politècnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola Enrgías Renovables, and by Grant N62909-09-1-4083 of the Office of Naval Research Global. At N.P.S., S.M., A.M., M.K., and F.X.G. were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. S.M., M.K., and A.M. are grateful to the National Research Council of the National Academies.

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Authors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, 833 Dyer Rd., SP249A, 93943, Monterey, CA, USA

    Simone Marras, Andreas Müller, Michal A. Kopera & Francis X. Giraldo

  2. Exa Corporation, Burlington, MA, USA

    James F. Kelly

  3. Barcelona Supercomputing Center BSC-CNS, CASE, Barcelona, Spain

    Margarida Moragues, Mariano Vázquez & Guillaume Houzeaux

  4. IIIA - CSIC, Bellaterra, Spain

    Mariano Vázquez

  5. Barcelona Supercomputing Center BSC-CNS, Earth Sciences, Barcelona, Spain

    Oriol Jorba

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Marras, S., Kelly, J.F., Moragues, M. et al. A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin. Arch Computat Methods Eng 23, 673–722 (2016). https://doi.org/10.1007/s11831-015-9152-1

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