Environmental Engineering Reference
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manner by controlling stability and treated the stiff parts of the system implicitly,
offer a promising opportunity to avoid splitting errors and perform larger time steps.
Furthermore, the advection part is usually integrated explicitly in time, where
the time step is constrained by a locally varying Courant-Friedrichs-Lewy (CFL)
number. In realistic scenarios there are usually regions of interest such as urban
areas which have to be examined more closely than surrounding regions. Therefore
the spatial grid in these regions is refined. Thus the smallest cell or more exactly
the cell with the smallest characteristic time determines the global advection time
step. Multirate schemes exploit the different time scales of the processes by using
different time steps for the subsystems. They can be employed to alter the time
step locally, leading to a significant reduction of computational cost. Such multirate
strategies have been developed since the 1980s when Osher and Sanders (1983)
introduced a simple multirate time integration scheme. Later, the approach was
refined by increasing the accuracy of multirate schemes based on explicit Runge-
Kutta methods (Constantinescu and Sandu, 2007). The generic recursive multirate
Runge-Kutta scheme RFSMR (“Recursive Flux Splitting Multirate”) has been
developed for the advection equation by Schlegel et al. (2009). It preserves the
linear invariants of the system and is of third order accuracy when applied to certain
explicit Runge-Kutta methods as base method. Furthermore, it can be easily adapted
to an arbitrary number of temporal refinement levels. The approach borrows ideas
from an IMEX splitting scheme introduced by Knoth and Wolke (1998), where
explicit Runge-Kutta methods are combined with an arbitrary implicit time integrator.
Due to the method properties, it may not only be applied to a splitting of spatial
domains with different time step restrictions, but also on a splitting of different
processes, e.g. advection, chemical reactions and aerosol dynamics.
In the paper, we combine the RFSMR approach with adapted IMEX schemes
for improving the numerical efficiency. An adaptive step size control as well as
the use of different explicit and implicit integration schemes is taken into account.
The accuracy and numerical efficiency of the implemented multirate methods are
analyzed for the coupled model system COSMO-MUSCAT (Wolke et al., 2004;
Hinneburg et al., 2009) and a set of selected artificial and real-life scenarios. In the
investigations, an extended version of the modal aerosol model M7 (Vignati et al.,
2004) is used for the treatment of aerosol processes. Furthermore, the gas phase
mechanisms RACM with 73 species and over 200 reactions is applied in this
study. The results are discussed in comparison to simulations using “operator
splitting” approaches.
2. The Multirate IMEX Approach
From the mathematical point of view, air quality models base on mass balances
which can be described by systems of time-dependent, three-dimensional advection-
diffusion-reaction equations (given in flux form)
 
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