Short Review of Current Numerical Developments in Meteorological Modelling
Authors: Steppeler Jürgen 
Published: July 2024

This paper reviews current numerical developments for atmospheric modelling. Numerical atmospheric modelling now looks back to a history of about 70 years after the first successful numerical prediction. Currently, we face new challenges, such as variable and adaptive resolution and ultra-highly resolving global models of 1 km grid length. Large eddy simulation (LES), special applications like the numerical prediction of pollution and atmospheric contaminants belong to the current challenges of numerical developments. While pollution prediction is a standard part of numerical modelling in case of accidents, models currently being developed aim at modelling pollution at all scales from the global to the micro scale. The methods discussed in this paper are spectral elements and other versions of Local-Galerkin (L-Galerkin) methods. Classic numerical methods are also included in the presentation. For example, the rather popular second-order Arakawa C-grid method can be shown to result as a special case of an L-Galerkin method using low-order basis functions. Therefore, developments for Galerkin methods also apply to this classic C-grid method, and this is included in this paper. The new generation of highly parallel computers requires new numerical methods, as some of the classic methods are not well suited for a high degree of parallel computing. It will be shown that some numerical inaccuracies need to be resolved and this indicates a potential for improved results by going to a new generation of numerical methods. The methods considered here are mostly derived from basis functions. Such methods are known under the names of Galerkin, spectral, spectral element, finite element or L-Galerkin methods. Some of these new methods are already used in realistic models. The spectral method, though highly used in the 1990s, is currently replaced by the mentioned local L-Galerkin methods. All methods presented in this review have been tested in idealized numerical situations, the so-called toy models. Waypoints on the way to realistic models and their mathematical problems will be pointed out. Practical problems of informatics will be highlighted. Numerical error traps of some current numerical approaches will be pointed out. These are errors not occurring with highly idealized toy models. Such errors appear when the test situation becomes more realistic. For example, many tests are for regular resolution and results can become worse when the grid becomes irregular. On the sphere no regular grids exist, except for the five derived from Platonic solids. Practical problems beyond mathematics on the way to realistic applications will also be considered. A rather interesting and convenient development is the general availability of computer power. For example, the computational power available on a normal personal computer is comparable to that of a supercomputer in 2005. This means that interesting developments, such as the small sphere atmosphere with a resolution of 1 km and a spherical circumference between 180 and 360 km are available to the normal owner of a personal computer (PC). Besides the mathematical problems of new approaches, we will also consider the informatics challenges of using the new generation of models on mainframe computers and PCs.