Need for Numerical Developments within COSMO
J. Steppeler, DWD Zurich 2005
• Klemp Wilhelmson• Runge Kutta • Semi-Lagrangian main Competitor of RK• Consideration in the medium range future: Global
nh
The development of LM numerics
• 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
Reasons for the success of KW
Grid Structure and Time Integration With Klemp Wilhelmson
Method
SFt
• 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
The Runge Kutta scheme (NCAR) t+dt
t
• 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
Considerations for RK
• 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
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
Vanished model Missing Numerical research
• 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
Considerations for COSMO
Global nh, the replacement of LM in the medium term
• Finite Volumes: 1
• Baumgardner Order2: 1
• Baumgardner Order3: 1
• Great circle grids: RK, SI, SL1 now 3 seem possible
• Tiled grids: 1.5
• Serendipidity grids 3
• Unstructured
• Conservation
Saving factors of Discretisations
3.11
1
2