1. Need for Numerical Developments within COSMO J
Need for Numerical Developments within COSMO J. Steppeler, DWD Zurich 2005

2. The development of LM numerics
Klemp Wilhelmson
Runge Kutta
Semi-Lagrangian main Competitor of RK
Consideration in the medium range future: Global nh

3. Reasons for the success of KW
Simple robust method comparable in cost and accuracy to Euler Centred difference
Competitive methods (SL) had (and have) problems in realising the efficiency gain they achieve in hydrostatic models as no efficient 3-d Helmholtz solver has yet been found.
With continued research into SI- methods this situation need not stay the same

4. Grid Structure and Time Integration With Klemp Wilhelmson Method

5. The Runge Kutta scheme (NCAR)
t+dt
RK is a two time level 3rd order in time scheme, involving substepping for fast waves
Spacial order is 3 or 5 (upstream differencing)
Approximation conditions concern vert. coordiante and phys. interface
Semi-lagrange: 2nd order in time, 3rd order in space, could be easier to achieve efficiency with large dt t

6. Considerations for RK
Accuracy 3rd order sufficient for practical purposes, if approximation conditions are fulfilled
Efficiency as KW
As a rather new method RK is not sufficiently investigated, in particular it would be nice to have more practical examples showing the advantages of the increased order in forecast mode

7. Numerical research Tends to be long term
Even for easy to program methods development time is often conted in years
Is often associated with model reprogramming
After lack of relevant developments models tend to stop existence

8. Vanished model Missing Numerical research
HIRLAM NH modelling
MC2 Further development of SL
eta Problems of Z
Tapp/White NH-instabilities
In institutes as NCAR or UKMO there is a planned change of model generations

9. Considerations for COSMO
Approximation conditions for RK scheme:
delta h < delta z or use LM_Z
Physics interfaceanalog to what is done for Orography
Increase of the efficiency of RK: Semi implicit methods e.t.c.
More validation of RK against other methods, such as KW, SL

10. Global nh, the replacement of LM in the medium term

11. Saving factors of Discretisations
Finite Volumes: 1
Baumgardner Order2: 1
Baumgardner Order3: 1
Great circle grids: RK, SI, SL 1 now 3 seem possible
Tiled grids:
Serendipidity grids 3
Unstructured
Conservation