1. COSMO-DE, M. Baldauf 18.09.2007 - 1 - Stability considerations of advection schemes and Improvements of Semi-Lagrange advection for the Runge-Kutta dynamical core Working Group 2: Numerics and Dynamics COSMO General Meeting 18.09.2006 Michael Baldauf michael.baldauf@dwd.de

2. COSMO-DE, M. Baldauf 18.09.2007 - 2 - up1 cd2 up3cd4up5cd6 Euler100000 LC-RK20.500.437000 LC-RK30.4190.5770.5420.4210.4780.364 LC-RK40.3480.7070.4360.5150.4330.446 LC-RK50.32200.39100.3290 LC-RK60.29600.38500.3110 LC-RK70.2820.2520.3690.1840.3230.159 Stability limit for the ‚effective Courant-number‘ for advection schemes C eff := C / s, s= stage of RK-scheme Baldauf (2007), submitted to J. Comput. Phys.

3. COSMO-DE, M. Baldauf 18.09.2007 - 3 - new: REAL FUNCTION sum_DDI( f(:,:) ) 1.) determine range of values: determine f min, f max globally 2.) number of quantisations: I Q = I max / N GP N GP = number of horizontal grid points (above 440000 at COSMO-EU) I max = 2 n-1 -1 n = 32 bit (4 Byte): I max = 2.1  10 9  I Q = 5000 (bad!) n = 64 bit (8 Byte): I max = 9.2  10 18  I Q = 2.1  10 13 (ok) (n = 96 bit (12 Byte): only user defined! seems not to be too difficult) 3.) Quantisation of field f  mapping to Integer values with quantisation steps  f = (f max - f min ) / I Q 4.) sum the Integers (Problem: MPI does not contain 8-Byte-Integer) 5.) back transformation to the REAL-number space accuracy: absolute error in the most unfavourable case:  S = | N GP *  f/2 | ~ (f max - f min )  N GP 2 / I max aim: Domain Decomposition Invariant sum