Meteoclim location:RMI Belgium date:5the november 2010
This abstract is about discretising mass-flux-type equations for the transport of conservative quantities in the
case that the quantity doesn't double in one timestep.
In the appendix of Geleyn et Al. 1982 (G82), a solution is presented seperatly for an downward resp. upward
direction of the flux, so where the direction doesn't change with height.
The discretization scheme is implicit, diagonal dominant and mass-conservative at the same time.
The issue at stake is to alter the algorithm so that it is also diagonal dominant in the case that the flux direction
changes with height. This is accomplished by looking at the way how the fluxes are discretisized in G82 when
the flux is upward on the one hand and downward on the other hand. The corresponding discretisizations for
each direction are applied and and are combined it into one discretization scheme. This way, the fluxes are
actually allways discretisized in the same way, but now taking into account his direction. Fortunatly, the scheme
is stable for every situation, whether or not the direction of the flux changes with height.
Another way of looking at the problem is to see the fluxes of different directions as seperate forcings, for which
we find a numerically discretisation algorithm independently. Perhaps this reasoning can be used in other
discretisation problems. Offcourse, one allways has to check the diagonal dominance for the case when
summing up the forcings after discretisation and before the matrix inversion of the implicit part. For the above
discretisation problem, this is fortunatly the case (allready mentioned above). Otherwise if it isn't the case, the
ultimate solution is to do the matrix inversion seperatly for each forcing (i.e. each direction), which of course
results in a different numerical result. In the latter, we actually apply the `operator splitting'-technique ref.
Fundamentals of atmospheric modeling (M. Jacobson).