Time Integration Techniques

The description of atmospheric processes generally leads to systems of stiff differential equations. For such systems explicit integration methods are only stable for very small time steps. For this purpose implicit-explicit (IMEX) methods were developed  [Bauer and Knoth, 2021; Knoth and Wolke, 1998]. The most important components of these IMEX methods are suitable algorithms for large systems of linear equations that make use of the special sparse structures of the problems [Knoth and Wensch, 2014; Soleimani et al., 2017]. A further gain in efficiency is obtained by so-called multi-rate procedures, which allow the use of different time step sizes in separate model regions and for different processes [Schlegel et al., 2012a; b].

 

 

Bauer, T. P., and O. Knoth (2021), Extended multirate infinitesimal step methods: Derivation of order conditions, J. Comput. Appl. Math., 387, 112541, doi:doi:10.1016/j.cam.2019.112541.

Knoth, O., and J. Wensch (2014), Generalized split-explicit Runge-Kutta methods for the compressible Euler equations, Mon. Wea. Rev., 142(5), 2067-2081, doi:doi:10.1175/MWR-D-13-00068.1.

Knoth, O., and R. Wolke (1998), Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows, Appl. Numer. Math., 28, 327-341.

Schlegel, M., O. Knoth, M. Arnold, and R. Wolke (2012a), Implementation of splitting methods for air pollution modeling, Geosci. Model Dev., 5(6), 1395-1405, doi:doi:10.5194/gmd-5-1395-2012.

Schlegel, M., O. Knoth, M. Arnold, and R. Wolke (2012b), Numerical solution of multiscale problems in atmospheric modeling, Appl. Numer. Math., 62(10), 1531-1543, doi:doi:10.1016/j.apnum.2012.06.023.

Soleimani, B., O. Knoth, and R. Weiner (2017), IMEX peer methods for fast-wave-slow-wave problems, Appl. Numer. Math., 118, 221-237, doi:10.1016/j.apnum.2017.02.016.

Computation of Atmospheric Flows on Cartesian Grids

Within various projects, the use of Cartesian grids for computing of atmospheric flows has been investigated. By using such grids, the stable atmosphere can be better described in structured terrain and an improvement in precipitation forecasts can be achieved. Block adaptive grids are used to obtain constant vertical resolutions. “Obstacles" (e.g. building structures) can also be embedded in these grids with the help of truncated cells. For this grid structure, adapted space and time discretisations were developed, implemented and tested within the flow model ASAM for different applications.