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