1. Improvement of the Semi-Lagrangian advection by ‘selective filling diffusion (SFD)' WG2-meeting COSMO-GM, Moscow, 06.09.2010 Michael Baldauf (FE13)

2. COSMO-Modell contains several methods for tracer advection: simple centered differences Lin, Rood-scheme In particular in combination with Runge-Kutta dynamical core: Bott-scheme (Finite Volume scheme) + locally conserving (at least for C<1) - direction splitting of 1D-steps  potential source of instabilities Semi-Lagrangian-scheme - not locally conserving + relatively robust - sometimes 'stripe patterns' along coordinate lines occur - in singular points high precipitation values can occur

3. COSMO-EU '02.05.2010' 0 UTC run 24h-precipitation sum SL with MF

4. COSMO-EU '02.05.2010' SL with SFD

5. advection eq. (1-dim.) rewritten as Semi-Lagrangian-Advection step 1: calculation of backward trajectory x j n-1 in principle any ODE-solver can be used (here: 2nd order) Staniforth, Côté (1991) MWR Baldauf, Schulz (2004) COSMO-Newsl. ~

6. 2nd step: Interpolation from neighbouring points linear weighting polynomials: cubic weighting polynomials: x,y,z  [0,1] = position in the grid cell (from backtrajectory calculation) q i,j,k = grid point value of q Semi-Lagrangian Advection i,j,k = -1,0 for tri-linear interpol.  8 grid points i,j,k = -2,...,1 for tri-cubic interpol.  64 grid points

7. properties of Semi-Lagrangian advection +unconditionally stable (i.e. no CFL condition, but Lifshitz-condition) +fully multi-dimensional scheme (no directional splitting necessary  quite robust) +increased efficiency if used for many tracers (calculation of backtrajectory only once) +linear scheme, if used without clipping + can be implemented also in unstructured grids + no non-linear instability if used for velocity advection -non-conserving scheme; but for higher order schemes conservation properties are not bad (without clipping): example: tri-cubic interpolation is exactly conserving in the case v=const (and cartesian grid) -multi-cubic interpolation  generates over-/undershoots  not positive definite for tracer advection: clipping of negative values necessary; this is a tremendous source of mass = strong violation of conservation (multi-linear interpolation  monotone, but highly diffusive)

8. 1D-Advection with v=const (CFL=0.6) exact solution cubic interpol. without clipping cubic interpol. with clipping cubic interpol. with SFD

9. FE 13 – 19.05.2015 Multiplicative Filling (Rood, 1987) SL - MF clipped values are globally summed and distributed over the whole field easy fast but only global conservation Problem of reproducibility: a sum of 'real' (=floating point) numbers is not associative: (a + b) + c  a + ( b + c ) solution: a sum of integer numbers is associative  map the Real number space to the Integer number space ( subroutine sum_DDI( field(:,:) ) in numeric_utilities_rk.f90 ) up to now: but this is an unsatisfying solution moreover on massively parallel computers: a global operation is needed

10. PBPV – 03/2010 to get closer to local conservation: fill negative values from positive values from the environment proposal: Semi-Lagrangian scheme with 'selective filling diffusion' (SFD) 1.tri-cubic interpolation 2.artificial 3D-diffusion only in the vicinity of negative values  fills up negative values diffusion itself can be formulated mass-conserving (FV) diffusion is ‘well-tempered’: only low requirements to the accuracy of the flux calculation,  relativiely efficient 3.if grid points with negative values remain  clipping

11. 1D-Advektion mit v=const (CFL=0.6) exact solution cubic interpol. without clipping cubic interpol. with clipping cubic interpol. with SFD

12. Idealised advection tests (with prescribed v-field) in the COSMO-Model Initialisierung '3D-Kegel-fkt.' in the following plots: difference against the analytic solution initial distribution: 3D-cone

13. SL - MFSL- SFD SL - clipBott Test 1: advection with v=const in terrain following grid (CFL=0.107)

14. PBPV – 03/2010 SL with Clipping: 5% mass increase! Bott: exactly conserving SL with 'SFD': 0.2% mass increase Test 1: advection with v=const in terrain following grid (CFL=0.107)

15. PBPV – 03/2010 SL with clipping: 2.7% mass increase! Bott: 0.1% mass increase SL with 'SFD': 0.15% mass increase Test 2: advection with v=const in terrain following grid (CFL=1.5

16. Test 3: Solid body rotation test  = (-3.5, -3.5, 280) * const (  1 turn around in 2 h) initial field: 3D-cone

17. Test 3: Solid body rotation test  = (-3.5, -3.5, 280) * const (  1 turn around in 2 h) SL - MFSL- SFD SL - clipBott

18. Test 3: Solid body rotation test  = (-3.5, -3.5, 280) * const (  1 turn around in 2 h) SL - MFSL- SFD SL - clipBott

19. SL with clipping: 8.5% mass increase! Bott: exactly conserving SL with 'SFD' 0.7% mass increase Conservation in the solid body rotation test

20. Test 4: 'LeVeque'-test (initial field: 3D-sphere) crashed SL - MFSL- SFD SL - clipBott

24. Synop-Verification: COSMO-EU (7km) 27.07.-27.08.2010 red: SL with SFD blue: SL with MF

25. Synop-Verification: COSMO-EU (7km) 27.07.-27.08.2010 red: SL with SFD blue: SL with MF

26. PBPV – 03/2010 Summary ‘selective filling diffusion (SFD)’ in the Semi-Lagrangian scheme improves local conservation properties (if non-negativeness is needed) often the 'best' scheme in idealised advection experiments ‘multiplicative filling’ no longer needed (but could be applied afterwards) improves linear properties of the tracer-advection synop-verification COSMO-EU (7km) (for 'August 2010'): small (but probably insignificant) improvements in RMSE slightly higher biases in general 'stripe-patterns' and tendency to spots with high precipitation has not improved outlook: some tuning of the SFD necessary (?) (Thresholds) Efficiency on vector computers (NEC SXx): 'diffusion in only a few points' ?  'diffusion everywhere with a lot K=0' ? tri-cubic interpolation not optimised for the NEC-SX9 (vectorisation degree is 99.8%, but a lot of bank conflicts)

28. Initialisation '3D-sphere'