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

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 e ciently 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 di erences, 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. (2003) [Review of numerical methods for nonhydrostatic weather prediction models Meteorol. Atmos. Phys. 82, 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.

Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

“…Recent reviews have pointed out the necessity to develop new-generation atmospheric models, using stateof-the-art numerical methods, to adequately capture all the physical processes needed for a complete representation of our climate system (Slingo et al 2009;Marras et al 2016). In particular, it has been argued that increasing the flexibility offered by the numerical grids used in atmospheric models will be crucial to improve the representation of the various spatial and physical scales involved (Williamson 2007;Slingo et al 2009;Staniforth and Thuburn 2012).…”

“…Advanced numerical methods are required for which the discretized equations can be formulated in a general framework while preserving important stability and accuracy properties. The numerical methods used to solve the basic flow equations are still often based on finite-difference methods, but if one wants to efficiently take advantage of advanced meshing techniques, numerical discretizations have to be redeveloped in consequence (Marras et al 2016). In this context, both finite-volume and finite-element methods (FEMs) emerge as good candidates to solve atmospheric flows on irregular, adaptive grids because of their overall flexibility, high scalability, and excellent conservation properties.…”

“…In this context, both finite-volume and finite-element methods (FEMs) emerge as good candidates to solve atmospheric flows on irregular, adaptive grids because of their overall flexibility, high scalability, and excellent conservation properties. In particular, FEM with inexact integration or mass lumping (including Galerkin and high-order spectral element methods) is becoming increasingly popular among atmospheric and climate modelers, and some of the most recent atmospheric solvers (both global and limited area) rely on such techniques (Giraldo et al 2002;Nair et al 2005;Giraldo and Restelli 2008;Nair et al 2009;Kelly and Giraldo 2012;Müller et al 2013;Kopera and Giraldo 2014;Marras et al 2015). However, Galerkin methods with exact integration and nondiagonal mass matrix have not previously been adopted, perhaps because of the extensive computational cost of solving the implicit system associated with them.…”

Simulation of all-scale atmospheric dynamics on unstructured meshes

“…Consequently, there is already available a substantial number of related reviews, developed from various angles and through varied methodological preferences, many of which are referenced in this paper. In particular, we refer the interested reader to [26,62] for indepth discussions of theories guiding multiscale atmospheric modelling and to [146,177,88,144,23,95], among the others, for specialised reviews of the numerical methods explored in dynamical cores of atmospheric models. To avoid duplicating the existing literature, we instead outline the historical background to the interdisciplinary crossfertilisation at the foundation of the all-scale atmospheric models, with particular emphasis on a select class of versatile finite-volume methods discussed in the body of the paper.…”

Numerical modelling of massif zonal structuring around underground working

“…Сущность метода состоит в том, что иско-мую непрерывную величину аппроксимируют кусоч-ным набором простейших функций, заданных над ограниченными конечными элементами, т.е. породный массив представляется в виде набора относительно больших конечных элементов, как правило, связанных между собой в отдельных узлах (Sdvyzhko, Babets, Kravchenko, & Smirnov, 2016;Marras et al, 2015).…”