Improvement of the Semi-Lagrangian advection by ‘selective filling diffusion (SFD)'

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

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

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

COSMO-EU '02.05.2010' SL with SFD

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

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

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)

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

up to now: • 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 ) but this is an unsatisfying solution moreover on massively parallel computers: a global operation is needed

to get closer to local conservation: • fill negative values from positive values from the environment • proposal: Semi-Lagrangian scheme with 'selective filling diffusion' (SFD) • tri-cubic interpolation • 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 • if grid points with negative values remain  clipping PBPV – 03/2010

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

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

Test 1: advection with v=const in terrain following grid (CFL=0.107) SL - MF SL- SFD SL - clip Bott

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

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

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

Test 3: Solid body rotation test  = (-3.5, -3.5, 280) * const ( 1 turn around in 2 h) SL - MF SL- SFD SL - clip Bott

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

Test 4: 'LeVeque'-test (initial field: 3D-sphere) SL - MF SL- SFD SL - clip Bott crashed

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

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 hasnot 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)

Initialisation '3D-sphere'

Improvement of the Semi-Lagrangian advection by ‘selective filling diffusion (SFD)'